1. If [tex]$A=\left[\begin{array}{ll}2 & 5 \\ 4 & 5\end{array}\right]$[/tex] and [tex]$B=\left[\begin{array}{ll}2 & 4 \\ 0 & 8\end{array}\right]$[/tex], then find [tex][tex]$A+B$[/tex][/tex] and [tex]$A-B$[/tex].

2. Let [tex]$A=\left[\begin{array}{ccc}3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4\end{array}\right]$[/tex] and [tex][tex]$B=\left[\begin{array}{ccc}7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12\end{array}\right]$[/tex][/tex], then find [tex]$5A+5B$[/tex].



Answer :

Certainly! Let's go through the questions one by one with detailed steps.

### Part 1: Finding [tex]\( A + B \)[/tex] and [tex]\( A - B \)[/tex]

Given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:

[tex]\[ A = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]

To find [tex]\( A + B \)[/tex]:

[tex]\[ A + B = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix} + \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]

We add corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:

[tex]\[ A + B = \begin{bmatrix} 2 + 2 & 5 + 4 \\ 4 + 0 & 5 + 8 \end{bmatrix} = \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \][/tex]

So, [tex]\( A + B \)[/tex] is:

[tex]\[ \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \][/tex]

To find [tex]\( A - B \)[/tex]:

[tex]\[ A - B = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix} - \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]

We subtract corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:

[tex]\[ A - B = \begin{bmatrix} 2 - 2 & 5 - 4 \\ 4 - 0 & 5 - 8 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \][/tex]

So, [tex]\( A - B \)[/tex] is:

[tex]\[ \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \][/tex]

### Part 2: Finding [tex]\( 5A + 5B \)[/tex]

Given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:

[tex]\[ A = \begin{bmatrix} 3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12 \end{bmatrix} \][/tex]

To find [tex]\( 5A + 5B \)[/tex]:

Firstly, calculate [tex]\( 5A \)[/tex] by multiplying all elements of matrix [tex]\( A \)[/tex] by 5.

[tex]\[ 5A = 5 \times \begin{bmatrix} 3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4 \end{bmatrix} = \begin{bmatrix} 15 & 5 & 10 \\ 0 & 40 & 35 \\ 45 & 55 & -20 \end{bmatrix} \][/tex]

Next, calculate [tex]\( 5B \)[/tex] by multiplying all elements of matrix [tex]\( B \)[/tex] by 5.

[tex]\[ 5B = 5 \times \begin{bmatrix} 7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12 \end{bmatrix} = \begin{bmatrix} 35 & 5 & 45 \\ 15 & 30 & -5 \\ 20 & -30 & 60 \end{bmatrix} \][/tex]

Now, add the resulting matrices [tex]\( 5A \)[/tex] and [tex]\( 5B \)[/tex]:

[tex]\[ 5A + 5B = \begin{bmatrix} 15 & 5 & 10 \\ 0 & 40 & 35 \\ 45 & 55 & -20 \end{bmatrix} + \begin{bmatrix} 35 & 5 & 45 \\ 15 & 30 & -5 \\ 20 & -30 & 60 \end{bmatrix} \][/tex]

Add corresponding elements:

[tex]\[ 5A + 5B = \begin{bmatrix} 15 + 35 & 5 + 5 & 10 + 45 \\ 0 + 15 & 40 + 30 & 35 + -5 \\ 45 + 20 & 55 + -30 & -20 + 60 \end{bmatrix} = \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \][/tex]

So, [tex]\( 5A + 5B \)[/tex] is:

[tex]\[ \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \][/tex]

In summary:

1. [tex]\( A + B = \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \)[/tex],
[tex]\( A - B = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \)[/tex].

2. [tex]\( 5A + 5B = \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \)[/tex].