# Matrix Formulas

Matrix Formulas

In mathematics the word ‘Matrix’ means  the rectangular array of numbers , symbols and expression. In order to know more about matrix click here.

Some important formulas of matrix are listed below:-

1. Transpose matrix A = $\begin{pmatrix}a & b \\ c & d \end{pmatrix}$ is a matrix then it’s transpose martis is

A’= $\begin{pmatrix}a & c \\ b & d\end {pmatrix}$

2. Zero matrix =(0,0), $\begin{pmatrix}0 &0 \\ 0 & 0 \end{pmatrix}$, etc.

3. unit matrix = $\begin{pmatrix}1 & 0 \\ 0 &1\end{pmatrix},\begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, etc.

4. Equal matrices; If A = $\begin{pmatrix}a & b \\ c & d \end{pmatrix}$ and B = $\begin {pmatrix} x & y \\ z & u \end{pmatrix}$

Then A = B if and only if a=x, b=y, c=z and d=u

i.e. corresponding elements of the two matrices are equal.

5. Addition and subraction of matrices: $A \pm B = \begin{pmatrix}a & b \\ c & d \end{pmatrix} \pm \begin{pmatrix}x & y \\ z & u\end{pmatrix} = \begin{pmatrix}a \pm x & b \pm y \\ c \pm z & d \pm u \end{pmatrix}$

6. Multiplication of a matrix by a vector: $(a \, b)_{1 \times 2} \begin{pmatrix}x & y \\ z & u \end{pmatrix} _{2 \times 2} = (ax + bz \, ay + bu ) _{1 \times 2}$

7. Multiplication of two matrices: $\begin{pmatrix}a & b \\ c & d \end{pmatrix} \begin{pmatrix} x & y \\ z & u \end {pmatrix} = \begin{pmatrix} ax + bz & ay + bu \\ cx + dz & cy + du \end{pmatrix}$

Note: Multiplication of two matrices exists if Number of row of first matrix is equal to number of column to another matrix..

8. Idempotent matrix: A square matrix A Is called idempotent if $A = A^2 = A^3 = \cdots$

9. A = $\begin{pmatrix}a & b \\ c & d \end{pmatrix}$ Determinant of A is $\begin{vmatrix}a & b \\ c & d \end{vmatrix} = (ad - bc)$

10. Inverse of matrix A = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $A ^{-1} = \dfrac{1}{|A|} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

11. Equation in matrix form: $\begin{pmatrix}a _1x + b_1y \\ a_2x +b_2y\end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$ Then $\begin{pmatrix}a _1 & b_1 \\ a_2 & b_2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$

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