Question

If \[ A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \] are three matrices, then find $ABC$.

Hint:

When multiplying three matrices, compute them two at a time: first $AB$, then multiply the result with $C$.

Answer 1

We are given matrices: \[ A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} \] Step 1: Compute $AB$ \[ AB = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix} \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix} \] Performing matrix multiplication: \[ AB = \begin{bmatrix} 1 \cdot 2 + (-1) \cdot (-1) + 0 \cdot 0 & 1 \cdot 0 + (-1) \cdot 3 + 0 \cdot 5 & 1 \cdot 1 + (-1) \cdot 4 + 0 \cdot 1 \end{bmatrix} = \begin{bmatrix} 2 + 1 + 0 & 0 - 3 + 0 & 1 - 4 + 0 \end{bmatrix} = \begin{bmatrix} 3 & -3 & -3 \end{bmatrix} \] Step 2: Compute $(AB)C$ \[ ABC = \begin{bmatrix} 3 & -3 & -3 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} = 3 \cdot 2 + (-3) \cdot 3 + (-3) \cdot 4 = 6 - 9 - 12 = -15 \]