Matrix Formulas





Matrix Formulas

In mathematics the word ‘Matrix’ meansĀ  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} ThenĀ  \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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