Question

find -4a + 4b
a = \

$$\begin{bmatrix}-7 & 7 \\\\ -8 & 4 \\\\ -8 & -5\\end{bmatrix}$$

\quad b = \

$$\begin{bmatrix}5 & 1 \\\\ -1 & 6 \\\\ 3 & -9\\end{bmatrix}$$

\textbf{a} \

$$\begin{bmatrix}8 & 24 \\\\ 28 & 8 \\\\ 20 & 16\\end{bmatrix}$$

\textbf{b} \

$$\begin{bmatrix}8 & 24 \\\\ 28 & 8 \\\\ -8 & -16\\end{bmatrix}$$

\textbf{c} \

$$\begin{bmatrix}48 & -24 \\\\ 28 & 8 \\\\ 44 & -16\\end{bmatrix}$$

\textbf{d} \

$$\begin{bmatrix}48 & -24 \\\\ -28 & -8 \\\\ 44 & -16\\end{bmatrix}$$

Explanation:

Step1: Factor out 4

We can factor out 4 from the expression \(-4A + 4B\) to get \(4(-A + B)=4(B - A)\). This simplifies the calculation by first finding the difference of matrices \(B\) and \(A\), then multiplying by 4.

Step2: Subtract matrix \(A\) from matrix \(B\)

For two matrices \(B=

$$\begin{bmatrix}5&1\\-1&6\\3&-9\end{bmatrix}$$

\) and \(A=

$$\begin{bmatrix}-7&7\\-8&4\\-8&-5\end{bmatrix}$$

\), the subtraction \(B - A\) is done element - wise.

• For the first row and first column: \(5-(-7)=5 + 7 = 12\)
• For the first row and second column: \(1 - 7=-6\)
• For the second row and first column: \(-1-(-8)=-1 + 8 = 7\)
• For the second row and second column: \(6 - 4 = 2\)
• For the third row and first column: \(3-(-8)=3 + 8 = 11\)
• For the third row and second column: \(-9-(-5)=-9 + 5=-4\)

So, \(B - A=

$$\begin{bmatrix}12&-6\\7&2\\11&-4\end{bmatrix}$$

\)

Step3: Multiply the resulting matrix by 4

Multiply each element of the matrix \(B - A\) by 4:

• First row, first column: \(4\times12 = 48\)
• First row, second column: \(4\times(-6)=-24\)
• Second row, first column: \(4\times7 = 28\)
• Second row, second column: \(4\times2 = 8\)
• Third row, first column: \(4\times11 = 44\)
• Third row, second column: \(4\times(-4)=-16\)

So, \(4(B - A)=

$$\begin{bmatrix}48&-24\\28&8\\44&-16\end{bmatrix}$$

\)

Answer:

C. \(

$$\begin{bmatrix}48&-24\\28&8\\44&-16\end{bmatrix}$$

\)