Exercise 3.1 · Q9 (MCQ): Which $x$ and $y$ make two matrices equal?

Basics you need

Solution

1. Same order, then equate entries

   Both matrices are $2\times2$⁠, so equality is possible in principle. Matching addresses:

   $$\begin{aligned} (1,1):&\quad 3x + 7 = 0 \\ (1,2):&\quad 5 = y - 2 \\ (2,1):&\quad y + 1 = 8 \\ (2,2):&\quad 2 - 3x = 4 \end{aligned}$$
   By Definition 2, the matrices are equal if and only if all four equations hold at the same time, for the same $x$ and the same $y$⁠.
   Equate entry $(i,j)$ with entry $(i,j)$⁠. Lesson: equality of matrices

2. The two equations for $y$ agree

   $$\begin{aligned} 5 = y - 2 &\;\Rightarrow\; y = 7 \\ y + 1 = 8 &\;\Rightarrow\; y = 7 \end{aligned}$$

   Both give $y = 7$⁠, so the $y$-entries can be matched.

   Add 2 to both sides of the first, subtract 1 from both sides of the second. The two requirements on $y$ are consistent.

3. The two equations for $x$ disagree

   $$\begin{aligned} 3x + 7 = 0 &\;\Rightarrow\; 3x = -7 \;\Rightarrow\; x = -\tfrac{7}{3} \\ 2 - 3x = 4 &\;\Rightarrow\; -3x = 2 \;\Rightarrow\; x = -\tfrac{2}{3} \end{aligned}$$

   The $(1,1)$ entry needs $x = -\frac73$⁠, but the $(2,2)$ entry needs $x = -\frac23$⁠. Since $-\frac73 \ne -\frac23$⁠, no value of $x$ satisfies both, and therefore no $x$⁠, $y$ make the matrices equal.

   Another way to see it: if both held, adding the equations $3x + 7 = 0$ and $2 - 3x = 4$ would give $9 = 4$⁠, which is false. So the four equations are inconsistent; the matrices are never equal, whatever $x$ and $y$ are.
   Two different required values for one unknown means no solution. Basics: solving equations

4. Examine every option

   Test each offered pair in all four entries of the first matrix and compare with $0, 5, 8, 4$ (the second matrix needs $y - 2 = 5$⁠).

   Option	Values	Verdict
   (A)	$x = -\frac13$⁠, $y = 7$	Wrong. $3x + 7 = -1 + 7 = 6 \ne 0$⁠, and $2 - 3x = 2 + 1 = 3 \ne 4$⁠.
   (B)	Not possible to find	Correct. Step 3 shows the $(1,1)$ and $(2,2)$ entries need different values of $x$⁠.
   (C)	$y = 7$⁠, $x = -\frac23$	Wrong. The $y$-entries and the $(2,2)$ entry match, but $3x + 7 = -2 + 7 = 5 \ne 0$⁠.
   (D)	$x = -\frac13$⁠, $y = -\frac23$	Wrong, and in every entry: $3x + 7 = -1 + 7 = 6 \ne 0$⁠, $2 - 3x = 2 + 1 = 3 \ne 4$⁠, $y - 2 = -\frac83 \ne 5$ and $y + 1 = \frac13 \ne 8$⁠.

   Each wrong option fails at least one entry, and one failing entry is enough to make the matrices unequal. Option (C) is the most tempting: it satisfies three of the four entries, but equality needs all four.

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