Matrices II

• Addition and Subtraction of 2 Matrices

2 matrices $A$ and $B$ can be added to or subtracted from each other ONLY WHEN THEIR ORDER IS SAME. These operations are performed element-wise. e.g.

${A = \begin {bmatrix} 1 & 2 & 3 \\ 9 & 6 & 4 \end {bmatrix}, B = \begin {bmatrix} 5 & 3 & 4 \\ 0 & 0 & 2 \end {bmatrix}}$

Then,

${A+B = \begin {bmatrix} 1+5 & 2+3 & 3+4 \\ 9+0 & 6+0 & 4+2 \end {bmatrix} = \begin {bmatrix} 6 & 5 & 7 \\ 9 & 6 & 6 \end {bmatrix}}$

${A-B = \begin {bmatrix} 1-5 & 2-3 & 3-4 \\ 9-0 & 6-0 & 4-2 \end {bmatrix} =\begin {bmatrix} -4 & -1 & -1 \\ 9 & 6 & 2 \end {bmatrix}}$

$A+B$ is always equal to $B+A$, but $A-B$ may not always be equal to $B-A$. We say matrix addition is commutative, but matrix subtraction is not commutative.

• Multiplication of 2 Matrices

The multiplication operation is unique to the matrices in the way it is defined. It has immense applications in various fields. Please note how it is performed:

When $B$ is to be postmultiplied with $A$, i.e. $A \times B$, the number of columns in $A$ must be equal to the number of rows in $B$.

Let ${A_{2 \times 3} = \begin {bmatrix} 1 & 2 & 3 \\ 9 & 6 & 4 \end {bmatrix}}$ and ${B_{3 \times 4} = \begin {bmatrix} 2 & 5 & -3 & 0 \\ 1 & 7 & -3 & 2 \\ -5 & 6 & 2 & 0 \end {bmatrix}}$.

Number of rows in $A$ = number of columns in $B$. Hence, $A \times B$ is possible and is given by:

${A \times B = \begin {bmatrix} 1(2)+ 2(1)+ 3(-5)& 1(5)+2(7)+3(6) & 1(-3)+2(-3)+3(2) & 1(0)+2(2)+3(0)\\ 9(2)+ 6(1)+ 4(-5)& 9(5)+6(7)+4(6) & 9(-3)+6(-3)+4(2) & 9(0)+6(2)+4(0) \end {bmatrix}}$

${A \times B = \begin {bmatrix} -11 & 37 & -3 & 4 \\ 4 & 111 & -37 & 18 \end {bmatrix}}$

Note that number of rows in $A \times B$ is equal to the number of rows in $A$ and number of columns in $A \times B$ is equal to the number of columns in $B$.

It is not possible to premultiply $A$ by $B$, i.e. $B \times A$, because number of columns in $B$ is NOT equal to number of rows in $A$.

• Elementary Transformations of a Matrix

A transformation transforms a matrix to another. There are 2 types of elementary transformations viz.

• Row Transformations

Consider ${A = \begin {bmatrix} 1 & 2 & 3 \\ 9 & 6 & 4 \end {bmatrix}}$.

1) ${R_{i}+ c{R_j}}$

Take $i$th row. To each element of $R_i$, add the number $c \times R_j$, such that both elements belong to same column. For example, $R_2 + 2 R_1$ would give,

${\begin {bmatrix} 1 & 2 & 3 \\ 9+2(1) & 6+2(2) & 4+2(3) \end {bmatrix} = \begin {bmatrix} 1 & 2 & 3 \\ 11 & 10 & 10 \end {bmatrix}}$

Call this matrix $D$.

2) ${c R_i}$

Take $i$th row. Multiply each element of $R_i$ by a constant $c$. For example, the matrix $D$ will get transformed to another matrix under $\frac 1 2 R_1$.

${\begin {bmatrix} \frac 1 2 & \frac 2 2 & \frac 3 2 \\ 11 & 10 & 10 \end {bmatrix} = \begin {bmatrix} 0.5 & 1 & 1.5 \\ 11 & 10 & 10 \end {bmatrix}}$

• Column Transformations

Again consider ${A = \begin {bmatrix} 1 & 2 & 3 \\ 9 & 6 & 4 \end {bmatrix}}$.

1) ${C_{i}+ c{C_j}}$

Take $i$th column. To each element of $C_i$, add the number $c \times C_j$, such that both elements belong to same row. For example, $C_2 + 2 C_1$ would give,

${\begin {bmatrix} 1 & 2+2(1) & 3 \\ 9 & 6+2(9) & 4 \end {bmatrix}= \begin {bmatrix} 1 & 4 & 3 \\ 9 & 24 & 4 \end {bmatrix}}$

Call this matrix $E$.

2) ${c C_i}$

Take $i$th column. Multiply each element of $C_i$ by a constant $c$. For example, the matrix $E$ will get transformed to another matrix under $\frac 1 4 C_3$.

${\begin {bmatrix} 1 & 4& \frac 3 4 \\ 9 & 24 & \frac 4 4 \end {bmatrix} = \begin {bmatrix} 1 & 4 & 0.75 \\ 9 & 24 & 1 \end {bmatrix}}$

NOTE: The transformations are a very useful tool in obtaining inverse of a matrix.