Question

If \(\begin{bmatrix}-1 & 3\\4 & -5\\0 & 2\end{bmatrix}\begin{bmatrix}1 & 2\\0 & 7\end{bmatrix}=\begin{bmatrix}-1 & 19\\\alpha & -27\\0 & 14\end{bmatrix}\), then the value of \(\alpha\) is

• A. 5
• B. 4
• C. 7
• D. -14
• E. -5

Hint:

Shortcut Tip: Never compute an entire matrix multiplication if you only need one element! Just find the intersection coordinates (row $i$, column $j$) of the unknown and compute that single dot product.

Answer 1

The Correct Option is B

Concept:
In matrix multiplication $AB = C$, the element $c_{ij}$ in the resulting matrix $C$ is found by taking the dot product of the $i$-th row of matrix $A$ and the $j$-th column of matrix $B$. We do not need to multiply the entire matrix to find a single missing variable; we just target the specific row and column.

Step 1: Locate the target variable $\alpha$.
Look at the resulting matrix on the right side of the equation: $$C = \begin{bmatrix}-1 & 19
\alpha & -27
0 & 14\end{bmatrix}$$ The variable $\alpha$ is located in the 2nd row, 1st column ($c_{21}$).

Step 2: Identify the corresponding row and column.
To find $c_{21}$, we must multiply the 2nd row of the first matrix by the 1st column of the second matrix. 2nd row of Matrix 1: $\begin{bmatrix}4 & -5\end{bmatrix}$ 1st column of Matrix 2: $\begin{bmatrix}1
0\end{bmatrix}$

Step 3: Set up the dot product equation.
Set the dot product of this row and column equal to $\alpha$: $$\alpha = (4 \times 1) + (-5 \times 0)$$

Step 4: Perform the multiplication.
Calculate the individual products: $$\alpha = (4) + (0)$$

Step 5: State the final answer.
Add the products together to find the value of $\alpha$: $$\alpha = 4$$ Hence the correct answer is (B) 4.