Here are four matrices: $\newline$ $\begin{array}{l} \mathbf{A}=\left[\begin{array}{l} 3 \\ 4 \\ 5 \end{array}\right] \quad \mathbf{B}=\left[\begin{array}{cc} -1 & -3 \\ 0 & 4 \end{array}\right] \\ \mathbf{C}=\left[\begin{array}{l} 2 \\ 1 \\ 0 \end{array}\right] \quad \mathbf{D}=\left[\begin{array}{ll} -5 & 2 \\ 3 & -4 \end{array}\right] \end{array}$ $\newline$ Which of the following calculations are possible? $\newline$ Choose all answers that apply: $\newline$ (A) $\mathbf{A}+\mathbf{C}$ $\newline$ (B) $\mathbf{B}+\mathbf{C}$ $\newline$ (C) $\mathbf{C}-\mathbf{A}$ $\newline$ (D) $\mathbf{D}-\mathbf{B}$

Full solution

Q. Here are four matrices:

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\begin{array}{l} \mathbf{A}=\left[\begin{array}{l} 3 \\ 4 \\ 5 \end{array}\right] \quad \mathbf{B}=\left[\begin{array}{cc} -1 & -3 \\ 0 & 4 \end{array}\right] \\ \mathbf{C}=\left[\begin{array}{l} 2 \\ 1 \\ 0 \end{array}\right] \quad \mathbf{D}=\left[\begin{array}{ll} -5 & 2 \\ 3 & -4 \end{array}\right] \end{array}

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Which of the following calculations are possible?

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Choose all answers that apply:

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(A)

\mathbf{A}+\mathbf{C}

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(B)

\mathbf{B}+\mathbf{C}

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(C)

\mathbf{C}-\mathbf{A}

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(D)

\mathbf{D}-\mathbf{B}

Check Matrix Dimensions: Matrix addition and subtraction are only possible when the matrices have the same dimensions. Let's check the dimensions of each matrix. $\newline$ Matrix A is a $3 \times 1$ matrix. $\newline$ Matrix B is a $2 \times 2$ matrix. $\newline$ Matrix C is a $3 \times 1$ matrix. $\newline$ Matrix D is a $2 \times 2$ matrix.
Check Matrix Operations: Now let's check each operation to see if it's possible by comparing the dimensions of the matrices involved. $\newline$ $A+C$ : Both $A$ and $C$ are $3 \times 1$ matrices, so this operation is possible.
Addition of A and C: $B+C$ : $B$ is a $2 \times 2$ matrix and $C$ is a $3 \times 1$ matrix. Since they have different dimensions, this operation is not possible.
Subtraction of B and C: $C-A$ : Both $C$ and $A$ are $3 \times 1$ matrices, so this operation is possible.
Subtraction of $C$ and $A$ : $D-B$ : Both $D$ and $B$ are $2 \times 2$ matrices, so this operation is possible.