Choose the operation and the matrix sizes (rows and columns), then enter the entries in the grids. The transpose Aᵀ and the power Aⁿ use matrix A only.
Enter the entries as numbers (decimals and negatives are fine). Addition and subtraction need A and B to be the same size; the product AB needs "columns of A = rows of B".
Result
Choose the operation and the matrix sizes on the left, enter the entries and press "Calculate". The resulting matrix and the steps will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formula
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Add \(A+B\), subtract \(A-B\) and multiply \(AB\) matrices from 2×2 to 4×4 on the spot (rectangular sizes such as 2×3 work too)
- You can also find the transpose \(A^{T}\), which swaps rows and columns, and powers \(A^{n}\) of a square matrix (\(n\) from 1 to 10)
- See how each calculation works, such as the "row × column" dot product in multiplication, in steps with your actual numbers
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets (MMULT, TRANSPOSE) and Python are all on this page
In the US, matrices usually first appear in Algebra 2 or Precalculus and are studied in depth in a college linear algebra course. Determinants and inverse matrices are not covered on this page.
What is this calculation used for?
Moving characters in games and 3D graphics
In 3D graphics, transformations such as rotating or scaling a character are calculated by "multiplying the coordinates by a matrix". Several transformations can be combined into one matrix by multiplying the matrices together first.
The GPU (graphics processor) in a game console or computer is a special-purpose calculator built to do huge numbers of these matrix multiplications.
Editing photos and images (an image is a matrix of numbers)
A digital image is itself a matrix: the brightness of each pixel arranged in rows and columns. Filters in photo apps, such as blur and sharpen, are calculated as operations on this matrix.
Seeing "an image as a matrix" is also the starting point of AI techniques such as image recognition.
Solving systems of equations at once (structures and circuits)
A system of equations with many unknowns can be written as a single equation, \(A\boldsymbol{x} = \boldsymbol{b}\), by putting the coefficients in a matrix.
In the strength analysis of buildings and the analysis of electric circuits, thousands or tens of thousands of equations are put into this form and solved at once by computer matrix calculations.
Counting routes in a transit map or links in a social network (adjacency matrix and powers)
Write the links between stations, or between people, as a matrix with "1 if connected, 0 if not" (an adjacency matrix). Then the entry in row \(i\), column \(j\) of the power \(A^{n}\) is "the number of ways to get from \(i\) to \(j\) by following exactly \(n\) links".
This idea is one of the foundations of route planners and of "friends of friends" analysis in social networks.
Estimating ripple effects in the economy (input-output tables)
National and regional economic statistics use a large matrix, called an input-output table, to show how much each industry buys from the others. The ripple effect, "if demand in one industry grows, how much does production grow in the others", is estimated with matrix calculations on this table (a method developed by Wassily Leontief, who won the Nobel Prize in Economics for it).
Inside AI (machine learning) is matrix multiplication
Most of the calculation in AI that handles text and images (neural networks) is repeating "multiply a list of input numbers by a learned matrix".
Because so much of the computing for training and running AI is matrix multiplication, GPUs, which are good at matrix calculations, have become essential for AI development.
Formula
Addition and subtraction (entry by entry in the same position)
Standard notation
(the usual math form)
\((A+B)_{ij}\)
\(=\)
\(a_{ij}\)
\(+\)
\(b_{ij}\)
In words
(symbols replaced with words)
③
entry in row \(i\), column \(j\) of the sum \(A+B\)
\(=\)
①
\(a_{ij}\): entry in row \(i\), column \(j\) of A
\(+\)
②
\(b_{ij}\): entry in row \(i\), column \(j\) of B
The formula in words
①
Add the
\(a_{ij}\): entry in row \(i\), column \(j\) of A
and the
②
\(b_{ij}\): entry in row \(i\), column \(j\) of B
, and you get the
③
entry in row \(i\), column \(j\) of the sum \(A+B\)
(for subtraction, just change "+" to "−")
Quick example
To add two 2×2 matrices, add the entries in the same position
\(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}\)
Key idea
You can add or subtract only when A and B have the same size (the same numbers of rows and columns). Addition gives the same answer in either order (\(A+B=B+A\)), but for subtraction, just as with ordinary numbers, the order changes the answer.
Multiplication (dot product of a row of A and a column of B)
Standard notation
(the usual math form)
\(c_{ij}\)
\(=\)
\(a_{i1}\)
\(\times\)
\(b_{1j}\)
\(+ \cdots +\)
\(a_{in}\)
\(\times\)
\(b_{nj}\)
In words
(symbols replaced with words)
③
\(c_{ij}\): entry in row \(i\), column \(j\) of the product \(AB\)
\(=\)
①
\(a_{i1}\): 1st entry in row \(i\) of A
\(\times\)
②
\(b_{1j}\): 1st entry in column \(j\) of B
\(+ \cdots +\)
\(a_{in}\): \(n\)th entry in row \(i\) of A
\(\times\)
\(b_{nj}\): \(n\)th entry in column \(j\) of B
The formula in words
①
Pair up the
\(a_{i1}, \dots, a_{in}\): entries in row \(i\) of A
with the
②
\(b_{1j}, \dots, b_{nj}\): entries in column \(j\) of B
in order from the start, multiply each pair and add them all, and you get the
③
\(c_{ij}\): entry in row \(i\), column \(j\) of the product \(AB\)
Quick example
The (1, 1) entry of AB is the dot product of "row 1 of A" and "column 1 of B". For example,
\(c_{11} = 1 \times 2 + 2 \times 1 = 4\)
\(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 1 \times 2 + 2 \times 1 & 1 \times 0 + 2 \times 2 \\ 3 \times 2 + 4 \times 1 & 3 \times 0 + 4 \times 2 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 10 & 8 \end{pmatrix}\)
Key idea
The product \(AB\) can be calculated (is defined) only when "columns of A = rows of B". The product of an \(m \times n\) matrix and an \(n \times p\) matrix is an \(m \times p\) matrix.
Also, unlike ordinary numbers, you cannot swap the order (\(AB\) and \(BA\) are generally not equal, and sometimes only one of them is defined).
Transpose (swap rows and columns)
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\((A^{T})_{ij}\)
\(=\)
\(a_{ji}\)
②
entry in row \(i\), column \(j\) of the transpose \(A^{T}\)
\(=\)
①
\(a_{ji}\): entry in row \(j\), column \(i\) of A
The formula in words
①
The
\(a_{ji}\): entry in row \(j\), column \(i\) of A
becomes, with rows and columns swapped, the
②
entry in row \(i\), column \(j\) of the transpose \(A^{T}\)
Quick example
Transposing a 2×3 matrix moves row 1 to column 1 and row 2 to column 2, giving a 3×2 matrix
\(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \quad \Longrightarrow \quad A^{T} = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}\)
Key idea
Besides \(A^{T}\), the transpose is also written \(A^{\top}\) or \(A'\) in some books and software. Transposing twice brings you back to the original (\((A^{T})^{T} = A\)).
Powers (multiplying the same square matrix again and again)
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(A^{n}\)
\(=\)
\(\underbrace{A\,A\,\cdots\,A}_{n}\)
②
\(A^{n}\): matrix \(A\) to the \(n\)th power
\(=\)
①
\(n\) copies of the square matrix \(A\) multiplied together
The formula in words
①
\(n\) copies of the square matrix \(A\) multiplied together
give
②
\(A^{n}\): matrix \(A\) to the \(n\)th power
(calculate in order from the left: \(A^{2}=AA\), \(A^{3}=A^{2}A\), and so on)
Quick example
The cube of \(A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) takes two multiplications:
\(A^{2} = AA = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
\(A^{3} = A^{2}A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}\)
Key idea
Powers exist only for square matrices (rows = columns). If the matrix is not square, the condition for the product \(AA\), "columns of A = rows of A", cannot be met.
Also note that you do not raise each entry to the \(n\)th power; you repeat matrix multiplication \(n-1\) times.
Matrix addition and subtraction work "entry by entry in the same position", and multiplication is built from "the dot product of a row of A and a column of B". Multiplication is defined only when "columns of A = rows of B", and the big difference from ordinary numbers is that swapping the order usually changes the result.
Symbols and terms
Symbols
| \(A,\ B\) | A, B | Names given to matrices. By custom, matrices are named with capital letters. |
| \(a_{ij}\) | a sub i j | The entry of matrix \(A\) in row \(i\), column \(j\) (the number in the \(i\)th row from the top and the \(j\)th column from the left). The small letters at the lower right (the subscripts) give its position. |
| \(b_{ij}\) | b sub i j | The entry of matrix \(B\) in row \(i\), column \(j\). It is read the same way as \(a_{ij}\); only the name of the matrix differs. |
| \(m \times n\) | m by n | The size (dimensions) of a matrix: \(m\) rows (horizontal lines) and \(n\) columns (vertical lines). (Example: a 2×3 matrix has 2 numbers down and 3 across, 6 numbers in all.) |
| \(c_{ij}\) | c sub i j | When the product \(AB\) is called \(C\), the entry of \(C\) in row \(i\), column \(j\). |
| \(A^{T}\) | A transpose | The transpose of \(A\): the matrix with rows and columns swapped. It is also written \(A^{\top}\) or \(A'\). |
| \(A^{n}\) | A to the n | The matrix made by multiplying \(n\) copies of \(A\). It is defined only for square matrices. |
| \(\sum\) | sigma | The sign for "add them all up". \(\sum_{k=1}^{n} a_{ik} b_{kj}\) tells you to add up \(a_{ik} b_{kj}\) for every \(k\) from 1 to \(n\). |
Terms
| matrix | Numbers arranged in a rectangle (rows and columns). Each number in it is called an entry. As in a spreadsheet, a position is given by "which row and which column". |
| row | A horizontal line of numbers in a matrix. Rows are counted from the top - row 1, row 2, and so on. Think of rows of seats in a theater. |
| column | A vertical line of numbers in a matrix. Columns are counted from the left - column 1, column 2, and so on. Think of the columns (pillars) of a building, which stand up and down. |
| entry | Each number in a matrix. It is also called an element. The entry in row \(i\), column \(j\) is written with subscripts, like \(a_{ij}\). |
| square matrix | A matrix with the same number of rows and columns (shaped like a square). Powers, as well as determinants and inverses (not covered on this page), exist only for square matrices. |
| transpose | The matrix with rows and columns swapped. The transpose of an \(m \times n\) matrix is an \(n \times m\) matrix. |
| dot product | The value you get by pairing up two lists of numbers from the start, multiplying each pair and adding them all. In matrix multiplication, the dot product of a row of A and a column of B gives one entry of the product. |
| linear algebra | The branch of mathematics that deals with matrices and vectors. It is often a required first- or second-year college course and is the foundation of computer graphics, statistics, AI and more. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| Operations with negative numbers (Grade 7) |
|
| Variables and subscripts (Grade 6 to high school) |
|
| Exponents (Grade 6) |
|
| Rows and columns of a table (elementary school) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the sum A + B
| Row 1 of matrix A | 1 | 2 |
| Row 2 of matrix A | 3 | 4 |
| Row 1 of matrix B | 5 | 6 |
| Row 2 of matrix B | 7 | 8 |
| Row 1 of A + B | =B1+B3 | =C1+C3 |
| Row 2 of A + B | =B2+B4 | =C2+C4 |
Table to find the difference A − B
| Row 1 of matrix A | 9 | 8 |
| Row 2 of matrix A | 7 | 6 |
| Row 1 of matrix B | 1 | 2 |
| Row 2 of matrix B | 3 | 4 |
| Row 1 of A − B | =B1-B3 | =C1-C3 |
| Row 2 of A − B | =B2-B4 | =C2-C4 |
Table to find the product AB
| Row 1 of matrix A | 1 | 2 |
| Row 2 of matrix A | 3 | 4 |
| Row 1 of matrix B | 2 | 0 |
| Row 2 of matrix B | 1 | 2 |
| AB (spills from row 1 automatically) | =MMULT(B1:C2,B3:C4) |
Table to find the transpose Aᵀ
| Row 1 of matrix A | 1 | 2 | 3 |
| Row 2 of matrix A | 4 | 5 | 6 |
| Aᵀ (spills from row 1 automatically) | =TRANSPOSE(B1:D2) |
Table to find the power A³
| Row 1 of matrix A | 1 | 1 |
| Row 2 of matrix A | 0 | 1 |
| A³ (spills from row 1 automatically) | =MMULT(MMULT(B1:C2,B1:C2),B1:C2) |
For addition and subtraction, just combine the cells in the same position, as in "=B1+B3" (in the first table, A + B comes out as 6, 8, 10, 12).
Use the MMULT function for multiplication and the TRANSPOSE function for the transpose. In recent Excel (such as Microsoft 365), the result spreads into the neighboring cells automatically (spill). In the third table AB is 4, 4, 10, 8, and in the fifth table A³ is 1, 3, 0, 1.
In older Excel, first select a range the same size as the result, type the formula, and confirm it with Ctrl+Shift+Enter.
Use the MMULT function for multiplication and the TRANSPOSE function for the transpose. In recent Excel (such as Microsoft 365), the result spreads into the neighboring cells automatically (spill). In the third table AB is 4, 4, 10, 8, and in the fifth table A³ is 1, 3, 0, 1.
In older Excel, first select a range the same size as the result, type the formula, and confirm it with Ctrl+Shift+Enter.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the sum A + B
| Row 1 of matrix A | 1 | 2 |
| Row 2 of matrix A | 3 | 4 |
| Row 1 of matrix B | 5 | 6 |
| Row 2 of matrix B | 7 | 8 |
| Row 1 of A + B | =B1+B3 | =C1+C3 |
| Row 2 of A + B | =B2+B4 | =C2+C4 |
Table to find the product AB
| Row 1 of matrix A | 1 | 2 |
| Row 2 of matrix A | 3 | 4 |
| Row 1 of matrix B | 2 | 0 |
| Row 2 of matrix B | 1 | 2 |
| AB (spills from row 1 automatically) | =MMULT(B1:C2,B3:C4) |
Table to find the transpose Aᵀ
| Row 1 of matrix A | 1 | 2 | 3 |
| Row 2 of matrix A | 4 | 5 | 6 |
| Aᵀ (spills from row 1 automatically) | =TRANSPOSE(B1:D2) |
Table to find the power A³
| Row 1 of matrix A | 1 | 1 |
| Row 2 of matrix A | 0 | 1 |
| A³ (spills from row 1 automatically) | =MMULT(MMULT(B1:C2,B1:C2),B1:C2) |
The same MMULT and TRANSPOSE functions as in Excel work as is. Copy the whole table and paste it into cell A1.
In Google Sheets, the results of MMULT and TRANSPOSE spread into the neighboring cells automatically (no Ctrl+Shift+Enter needed).
In Google Sheets, the results of MMULT and TRANSPOSE spread into the neighboring cells automatically (no Ctrl+Shift+Enter needed).
How to calculate it in Python
A = [[1, 2], [3, 4]]
B = [[2, 0], [1, 2]]
def matrix_add(x, y): # addition (add the entries in the same position)
return [[x[i][j] + y[i][j] for j in range(len(x[0]))] for i in range(len(x))]
def matrix_multiply(x, y): # multiplication (dot product of a row of A and a column of B)
return [[sum(x[i][k] * y[k][j] for k in range(len(y))) for j in range(len(y[0]))] for i in range(len(x))]
def matrix_transpose(x): # transpose (swap rows and columns)
return [[x[i][j] for i in range(len(x))] for j in range(len(x[0]))]
def matrix_power(x, n): # power (multiply n times in order from the left)
result = x
for _ in range(n - 1):
result = matrix_multiply(result, x)
return result
print("A + B =", matrix_add(A, B))
print("AB =", matrix_multiply(A, B))
print("Transpose of A =", matrix_transpose(A))
print("A cubed =", matrix_power(A, 3))
Runs with the standard library only. Replace A and B at the top with your own matrices and run it (for subtraction, just change the "+" in matrix_add to "-"). For serious numerical work, the standard choice is NumPy, a library made for matrices (multiplication can be written "A @ B").
How to write it in LaTeX and other math languages (copy and paste)
Addition and subtraction (entry by entry in the same position)
(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ
(A + B)_{ij} = a_{ij} + b_{ij}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub>
<mrow><mo>(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo>)</mo></mrow>
<mrow><mi>i</mi><mi>j</mi></mrow>
</msub>
<mo>=</mo>
<msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub>
<mo>+</mo>
<msub><mi>b</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub>
</mrow>
</math>
(A + B)_(ij) = a_(ij) + b_(ij)
A + B
C := A + B;
C = A + B;
(A + B)_ij = a_ij + b_ij
Multiplication (dot product of a row of A and a column of B)
cᵢⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + ⋯ + aᵢₙbₙⱼ
c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>c</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub>
<mo>=</mo>
<munderover>
<mo>∑</mo>
<mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow>
<mi>n</mi>
</munderover>
<msub><mi>a</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub>
<msub><mi>b</mi><mrow><mi>k</mi><mi>j</mi></mrow></msub>
</mrow>
</math>
c_(ij) = sum_(k=1)^(n) a_(ik) b_(kj)
A . B
C := A . B;
C = A * B;
c_ij = ∑_(k=1)^n a_ik b_kj
Transpose (swap rows and columns)
(Aᵀ)ᵢⱼ = aⱼᵢ
(A^{T})_{ij} = a_{ji}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub>
<mrow><mo>(</mo><msup><mi>A</mi><mi>T</mi></msup><mo>)</mo></mrow>
<mrow><mi>i</mi><mi>j</mi></mrow>
</msub>
<mo>=</mo>
<msub><mi>a</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub>
</mrow>
</math>
(A^T)_(ij) = a_(ji)
Transpose[A]
C := LinearAlgebra:-Transpose(A);
C = A.';
(A^T)_ij = a_ji
Powers (multiplying the same square matrix again and again)
Aⁿ = AA⋯A
A^{n} = \underbrace{AA\cdots A}_{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mi>A</mi><mi>n</mi></msup>
<mo>=</mo>
<mi>A</mi><mi>A</mi><mo>⋯</mo><mi>A</mi>
</mrow>
</math>
A^n
MatrixPower[A, n]
C := A^n;
C = A^n;
A^n
How to have ChatGPT do the calculation
You are a calculation assistant for matrices. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the matrices A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 2]], find each of the following matrices. 1. The sum A + B 2. The difference A − B 3. The product AB 4. The transpose of A 5. A cubed (A to the 3rd power) For each one, show the formula you used (which entries you combined and how) and the resulting matrix from the execution.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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