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Fill in the missing number in the equation:

[tex]\[
3\left[\begin{array}{cc}
-1 & 2 \\
4 & -5
\end{array}\right]+5\left[\begin{array}{cc}
-1 & 2 \\
4 & -5
\end{array}\right]=\square \left[\begin{array}{cc}
-1 & 2 \\
4 & -5
\end{array}\right]
\][/tex]



Answer :

To find the missing number in the equation, we need to follow these steps:

Step 1: Multiply the matrix by 3 and 5 respectively.

Given matrix:
[tex]\[ \begin{bmatrix} -1 & 2 \\ 4 & -5 \end{bmatrix} \][/tex]

First, multiply by 3:
[tex]\[ 3 \times \begin{bmatrix} -1 & 2 \\ 4 & -5 \end{bmatrix} = \begin{bmatrix} 3 \times -1 & 3 \times 2 \\ 3 \times 4 & 3 \times -5 \end{bmatrix} = \begin{bmatrix} -3 & 6 \\ 12 & -15 \end{bmatrix} \][/tex]

Next, multiply by 5:
[tex]\[ 5 \times \begin{bmatrix} -1 & 2 \\ 4 & -5 \end{bmatrix} = \begin{bmatrix} 5 \times -1 & 5 \times 2 \\ 5 \times 4 & 5 \times -5 \end{bmatrix} = \begin{bmatrix} -5 & 10 \\ 20 & -25 \end{bmatrix} \][/tex]

Step 2: Add the resulting matrices.

Add:
[tex]\[ \begin{bmatrix} -3 & 6 \\ 12 & -15 \end{bmatrix} + \begin{bmatrix} -5 & 10 \\ 20 & -25 \end{bmatrix} = \begin{bmatrix} -3 + -5 & 6 + 10 \\ 12 + 20 & -15 + -25 \end{bmatrix} = \begin{bmatrix} -8 & 16 \\ 32 & -40 \end{bmatrix} \][/tex]

Step 3: Determine the scalar multiple.

The resulting matrix [tex]\(\begin{bmatrix} -8 & 16 \\ 32 & -40 \end{bmatrix} \)[/tex] is a scalar multiple of the original matrix [tex]\(\begin{bmatrix} -1 & 2 \\ 4 & -5 \end{bmatrix} \)[/tex].

To find the scalar, we compare the corresponding elements of the two matrices:
[tex]\[ \frac{-8}{-1} = 8, \quad \frac{16}{2} = 8, \quad \frac{32}{4} = 8, \quad \frac{-40}{-5} = 8 \][/tex]

In all cases, the scalar multiple is [tex]\( 8 \)[/tex].

Therefore, the missing number in the equation is [tex]\( 8 \)[/tex]:

[tex]\[ 3\left[\begin{array}{cc} -1 & 2 \\ 4 & -5 \end{array}\right] + 5\left[\begin{array}{cc} -1 & 2 \\ 4 & -5 \end{array}\right] = 8 \cdot \left[\begin{array}{cc} -1 & 2 \\ 4 & -5 \end{array}\right] \][/tex]