Question

If \(\mathbf{A}=\left(\begin{array}{rrr}{3} & {2} & {-1} \\ {2} & {-1} & {2} \\\ {1} & {2} & {1}\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr}{2} & {1} & {-1} \\ {-2} & {3} & {3} \\\ {1} & {0} & {2}\end{array}\right),\) verify that \(2(\mathbf{A}+\mathbf{B})=2 \mathbf{A}+2 \mathbf{B}\)

Short answer

Answer: Yes

Step-by-step solution

Understand the given matrices and equation

We have two matrices A and B: \(\mathbf{A}=\left(\begin{array}{rrr} {3} & {2} & {-1} \\ {2} & {-1} & {2} \\ {1} & {2} & {1} \end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr} {2} & {1} & {-1} \\ {-2} & {3} & {3} \\ {1} & {0} & {2} \end{array}\right)\) We have to verify the following equation treating both sides as separate calculations: \(2(\mathbf{A}+\mathbf{B})=2 \mathbf{A}+2 \mathbf{B}\)

Calculate \(\mathbf{A}+\mathbf{B}\)

To add two matrices, we add the corresponding elements in each matrix: $\mathbf{A}+\mathbf{B}=\left(\begin{array}{rrr} {3+2} & {2+1} & {-1+(-1)} \\ {2+(-2)} & {-1+3} & {2+3} \\ {1+1} & {2+0} & {1+2} \end{array}\right) =\left(\begin{array}{rrr} {5} & {3} & {-2} \\ {0} & {2} & {5} \\ {2} & {2} & {3} \end{array}\right)$

Calculate \(2(\mathbf{A}+\mathbf{B})\)

To perform scalar multiplication, we multiply each element of the given matrix by the scalar value: $2(\mathbf{A}+\mathbf{B})=2\left(\begin{array}{rrr} {5} & {3} & {-2} \\ {0} & {2} & {5} \\ {2} & {2} & {3} \end{array}\right) =\left(\begin{array}{rrr} {10} & {6} & {-4} \\ {0} & {4} & {...