ADDITION AND SUBTRACTION OF MATRIX

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A matrix is a rectangular array of numbers.

Two matrixes can be added or subtracted if they have the same dimensions.

For example, We can add a 2 x 2 matrix to another 2 x 2 matrix.

But, we can not add or subtract 2 x 3 matrix to another 3 x3 matrix.

The procedure for adding matrix is exemplified below

$$\left[{\begin{array}{cc} A & B \\ C & D  \\ \end {array} } \right]+\left[{\begin{array}{cc} E & F\\ G & H\\ \end {array} } \right]=\left[{\begin{array}{cc} A+E & B+F \\ C+G & D+H\\ \end {array} } \right]$$

The procedure for subtracting matrix is exemplified below

$\left[{\begin{array}{cc} A & B \\ C & D  \\ \end {array} } \right]-\left[{\begin{array}{cc} E & F\\ G & H\\ \end {array} } \right]=\left[{\begin{array}{cc} A-E & B-F \\ C-G & D-H\\ \end {array} } \right]$

Example 1

If $A=\left[{\begin{array}{cc} -3 & 0 \\ 7 & -4\\ \end {array} } \right]$, $B=\left[{\begin{array}{cc} -1 & 0 \\ -2 & 4\\ \end {array} } \right]$

Find

1. A+B

2. A-B

Solution:

1. $A+B=\left[{\begin{array}{cc} -3  & 0 \\ 7 & -4 \\ \end {array} } \right]+\left[{\begin{array}{cc} -1 & 0\\ -2 & 4 \\ \end {array} } \right]$

$A+B=\left[{\begin{array}{cc} -3+(-1) & 0+0 \\ 7+(-2) & -4 +4 \\ \end {array} } \right]$

$A+B=\left[{\begin{array}{cc} -4 & 0 \\ 5 & 0 \\ \end {array} } \right]$

2. $A-B=\left[{\begin{array}{cc} -3  & 0 \\ 7 & -4 \\ \end {array} } \right]-\left[{\begin{array}{cc} -1 & 0\\ -2 & 4 \\ \end {array} } \right]$

$A-B=\left[{\begin{array}{cc} -3-1 & 0-0 \\ 7-(-2) & -4 -4 \\ \end {array} } \right]$

$A-B=\left[{\begin{array}{cc} -4 & 0 \\ 9 & -8 \\ \end {array} } \right]$

Example 2

If $A=\left[{\begin{array}{cccc} 4 & 3 \\ 6 & 8 \\  9 & 5 \\ \end {array} } \right]$, $B=\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\  6 & 5 \\ \end {array} } \right]$

Determine

1. A+B

2. A+A

3. B+B

4. B-A

5. A-B

6. B-B

7. Is BB a zero matrix 

Solution:

1. $A+B=\left[{\begin{array}{cccc} 4 & 3 \\ 6 & 8 \\ 9 & 5 \end {array} } \right]+\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]$

$A+B=\left[{\begin{array}{cccc} 4+3 & 3+4 \\ 6+2 & 8+1 \\ 9+6 & 5+5 \end {array} } \right]$

$A+B=\left[{\begin{array}{cccc} 7 & 7 \\ 8 & 9 \\ 15 & 10 \end {array} } \right]$

2. $A+A=\left[{\begin{array}{cccc} 4 & 3 \\ 6 & 8 \\ 9 & 5 \end {array} } \right]+\left[{\begin{array}{cccc} 4 & 3 \\ 6 & 8 \\ 9 & 5 \end {array} } \right]$

$A+A=\left[{\begin{array}{cccc} 4+4 & 3+3 \\ 6+6 & 8+8 \\ 9+9 & 5+5 \end {array} } \right]$

$A+A=\left[{\begin{array}{cccc} 8 & 6 \\ 12 & 16 \\ 18 & 10 \end {array} } \right]$

3. $B+B=\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]+\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]$

$B+B=\left[{\begin{array}{cccc} 3+3 & 4+4 \\ 2+2 & 1+1 \\ 6+6 & 5+5 \end {array} } \right]$

$B+B=\left[{\begin{array}{cccc} 6 & 8 \\ 4 & 2 \\ 12 & 10 \end {array} } \right]$

4. $B-A=\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]-\left[{\begin{array}{cccc} 4 & 3 \\ 6 & 8 \\ 9 & 5 \end {array} } \right]$

$B-A=\left[{\begin{array}{cccc} 3-4 & 4-3 \\ 2-6 & 1-8 \\ 6-9 & 5-5 \end {array} } \right]$

$B-A=\left[{\begin{array}{cccc} -1 & 1 \\ -4 & -7 \\ -3 & 0 \end {array} } \right]$

5. $A-B=\left[{\begin{array}{cccc} 4 & 3 \\ 6 & 8 \\ 9 & 5 \end {array} } \right]-\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]$

$A-B=\left[{\begin{array}{cccc} 4-3 & 3-4 \\ 6-2 & 8-1 \\ 9-6 & 5-5 \end {array} } \right]$

$A-B=\left[{\begin{array}{cccc} 1 & -1 \\ 4 & 7 \\ 3 & 0 \end {array} } \right]$

6. $B-B=\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]-\left[{\begin{array}{cccc} 3 & 4 \\ 2 & 1 \\ 6 & 5 \end {array} } \right]$

$B-B=\left[{\begin{array}{cccc} 3-3 & 4-4 \\ 2-2 & 1-1 \\ 6-6 & 5-5 \end {array} } \right]$

$B-B=\left[{\begin{array}{cccc} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end {array} } \right]$

7. As you can see, $B-B$ has zero in all its entries. Hence, it is a zero matrix.

A zero matrix is a matrix that has all it's entries equal to zero.

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