Question

If $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & a \\ 2 & 4 & 7 \end{bmatrix}$ and $B = \begin{bmatrix} 13 & 2 & b \\ -3 & -1 & 2 \\ -2 & 0 & 1 \end{bmatrix}$ where matrix $B$ is the inverse of matrix $A$, then the values of $a$ and $b$ are

• A. $a = -5, b = 7$
• B. $a = 7, b = -5$
• C. $a = -7, b = 5$
• D. $a = 5, b = -7$

Hint:

When a matrix equation involves the identity condition $A \cdot A^{-1} = I$, you don't need to compute the full inverse or expand the entire matrix multiplication! Pick rows and columns strategically to solve for each unknown variable independently in a single step.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given two $3 \times 3$ matrices, $A$ and $B$, where $B = A^{-1}$. We need to find the values of the unknown scalar variables $a$ and $b$.

Step 2: Key Formula or Approach:
Since matrix $B$ is the inverse of matrix $A$, their product must yield the identity matrix $I$: $$A \cdot B = I \implies \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & a \\ 2 & 4 & 7 \end{bmatrix} \begin{bmatrix} 13 & 2 & b \\ -3 & -1 & 2 \\ -2 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ We can determine $a$ and $b$ directly by performing row-by-column multiplication for specific elements that isolate the parameters.

Step 3: Detailed Explanation:
Let's find a product entry that involves only one unknown parameter to keep things simple:

• Finding $a$: Look at the element in Row 3, Column 3 of the identity matrix, which must equal 1. This entry is formed by multiplying Row 3 of matrix $A$ by Column 3 of matrix $B$: $$(2)(b) + (4)(2) + (7)(1) = 1 \implies 2b + 8 + 7 = 1 \implies 2b + 15 = 1$$ $$2b = 1 - 15 = -14 \implies b = -7$$

• Finding $b$: Now look at the element in Row 1, Column 3 of the identity matrix, which must equal 0. This entry is formed by multiplying Row 1 of matrix $A$ by Column 3 of matrix $B$: $$(1)(b) + (2)(2) + (3)(1) = 0$$ Substitute $b = -7$ to check consistency: $$-7 + 4 + 3 = 0 \implies 0 = 0$$ This is consistent. Now, let's look at an entry containing $a$, such as Row 2, Column 1 of the identity matrix, which must equal 0. Multiply Row 2 of matrix $A$ by Column 1 of matrix $B$: $$(1)(13) + (1)(-3) + (a)(-2) = 0$$ $$13 - 3 - 2a = 0 \implies 10 - 2a = 0$$ $$2a = 10 \implies a = 5$$
This gives our final values: $a = 5$ and $b = -7$.

Step 4: Final Answer:
The values of $a$ and $b$ are $5$ and $-7$ respectively, which corresponds to option (D).