See also: Gauss-Jordan Elimination, Simultaneous Linear Equations, Geometric Linear Transformation

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A matrix is a rectangular array of numbers.

The size of a matrix is its dimension, namely the number of rows and columns of the matrix.

For operations of matrices, please use the two calculators below.

• Matrix Multiplication, Addition and Subtraction Calculator
• Matrix Inverse, Determinant and Adjoint Calculator

To find inverse of matrix, you can also use the Gauss-Jordan Elimination method.

Read explanation about matrix operations below.

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Matrix Operations

Addition and Subtraction of Matrices

If matrices $A$ and $B$ are of the same size,

• the sum $A+B$ is the matrix obtained by adding the entries of $B$ to the corresponding entries of $A$.
• the difference $A-B$ is the matrix obtained by subtracting the entries of $B$ from the corresponding entries of $A$.

If $A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \text{…}& a_{1n}\\ a_{21}& a_{22}& \text{…}& a_{2n}\\ \text{⋮}& \text{⋮}& \text{⋱}& \text{⋮}\\ a_{m1}& a_{m2}& \text{…}& a_{mn}\end{array}\right)$ and $B=\left(\begin{array}{cccc}{b}_{11}& b_{12}& \text{…}& a_{1n}\\ b_{21}& b_{22}& \text{…}& a_{2n}\\ \text{⋮}& \text{⋮}& \text{⋱}& \text{⋮}\\ b_{m1}& b_{m2}& \text{…}& b_{mn}\end{array}\right)$

$A+B=\left(\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& \text{…}& a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& \text{…}& a_{2n}+b_{2n}\\ \text{⋮}& \text{⋮}& \text{⋱}& \text{⋮}\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& \text{…}& a_{mn}+b_{mn}\end{array}\right)$

$A-B=\left(\begin{array}{cccc}{a}_{11}-b_{11}& a_{12}-b_{12}& \text{…}& a_{1n}-b_{1n}\\ a_{21}-b_{21}& a_{22}-b_{22}& \text{…}& a_{2n}-b_{2n}\\ \text{⋮}& \text{⋮}& \text{⋱}& \text{⋮}\\ a_{m1}-b_{m1}& a_{m2}-b_{m2}& \text{…}& a_{mn}-b_{mn}\end{array}\right)$

Matrices of different sizes cannot be added or subtracted.

Example:

If $A=\left(\begin{array}{cc}1& 2\\ 0& -3\end{array}\right)$ and $B=\left(\begin{array}{cc}3& 1\\ -1& 2\end{array}\right)$

$A+B=\left(\begin{array}{cc}1& 2\\ 0& -3\end{array}\right)+\left(\begin{array}{cc}3& 1\\ -1& 2\end{array}\right)=\left(\begin{array}{cc}4& 3\\ -1& -1\end{array}\right)$

$A-B=\left(\begin{array}{cc}1& 2\\ 0& -3\end{array}\right)-\left(\begin{array}{cc}3& 1\\ -1& 2\end{array}\right)=\left(\begin{array}{cc}-2& 1\\ 1& -5\end{array}\right)$

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Multiplication of Matrices

If $A$ is an $m\times r$ matrix and $B$ is an $r\times n$ matrix, the product $A<!--\; InvisibleTimes\; -->B$ is an $m\times n$ matrix whose entry from row $i$ and column $j$ is the sum of the products of the corresponding entries from row $i$ of $A$ and column $j$ of $B$.

The entry ${\left(AB\right)}_{ij}$ in row $i$ and column $j$ of $A<!--\; InvisibleTimes\; -->B$ is given by

$${\left(AB\right)}_{ij}=a_{i1}<!--\; InvisibleTimes\; -->b_{1j}+a_{i2}<!--\; InvisibleTimes\; -->b_{2j}+\text{…}+a_{ir}<!--\; InvisibleTimes\; -->b_{rj}$$

Matrices $A$ and $B$ can only be multiplied if the number of columns of $A$ is the same as the number of rows of $B$.

Example:

$A=\left(\begin{array}{ccc}1& 2& 1\\ 0& -3& 2\end{array}\right)$ and $B=\left(\begin{array}{cccc}3& 1& 0& 1\\ -1& 2& 3& 0\\ 0& -2& 1& 1\end{array}\right)$

$A<!--\; InvisibleTimes\; -->B=\left(\begin{array}{ccc}1& 2& 1\\ 0& -3& 2\end{array}\right)<!--\; InvisibleTimes\; -->\left(\begin{array}{cccc}3& 1& 0& 1\\ -1& 2& 3& 0\\ 0& -2& 1& 1\end{array}\right)=\left(\begin{array}{cccc}1& 3& 7& 2\\ 3& -10& -7& 2\end{array}\right)$

• The element at row 1 and column 1 of $A<!--\; InvisibleTimes\; -->B$ is obtained from summing up the product of corresponding entries of row 1 of $A$ and column 1 of $B$, i.e.
  ${\left(AB\right)}_{11}=\left(1\right)\left(3\right)+\left(2\right)\left(-1\right)+\left(1\right)\left(0\right)=1$
• The element at row 1 and column 2 of $A<!--\; InvisibleTimes\; -->B$ is obtained from summing up the product of corresponding entries of row 1 of $A$ and column 2 of $B$, i.e.
  ${\left(AB\right)}_{12}=\left(1\right)\left(1\right)+\left(2\right)\left(2\right)+\left(1\right)\left(-2\right)=3$
• The element at row 2 and column 1 of $A<!--\; InvisibleTimes\; -->B$ is obtained from summing up the product of corresponding entries of row 2 of $A$ and column 1 of $B$, i.e.
  ${\left(AB\right)}_{21}=\left(0\right)\left(3\right)+\left(-3\right)\left(-1\right)+\left(2\right)\left(0\right)=3$
• And so on

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Inverse of a Matrix

The inverse of a square matrix $A$ is the matrix $A^{-1}$ such that $A{A}^{-1}=I$

Example:

If $A=\left(\begin{array}{cc}-3& 2\\ 5& -4\end{array}\right)$, then $A^{-1}=\left(\begin{array}{cc}-2& -1\\ -2.5& -1.5\end{array}\right)$

because $A{A}^{-1}=\left(\begin{array}{cc}-3& 2\\ 5& -4\end{array}\right)\left(\begin{array}{cc}-2& -1\\ -2.5& -1.5\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

One way to get the inverse of a square matrix $A$ is to use the following formula

$$A^{-1}=\frac{\mathrm{adj}\left(A\right)}{\det \left(A\right)}$$

If the determinant of the matrix is 0, the matrix doesn't have an inverse and it's called a singular matrix.

Another way to find the inverse of a matrix is to append an identity matrix on the right side of the matrix then use the Gauss-Jordan Elimination method to reduce the matrix to its reduced row echelon form.

Confused and have questions? We’ve got answers. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field.

By Jimmy Sie

See also: Gauss-Jordan Elimination, Simultaneous Linear Equations, Geometric Linear Transformation