Q.Which of the given values of $x$ and $y$ make the following pair of matrices equal $\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}$, $\begin{bmatrix} 0 & y-2 \\ 8 & 4 \end{bmatrix}$ (A) $x = -\frac{1}{3}$, $y=7$ (B) Not possible to find (C) $y=7$, $x=-\frac{2}{3}$ (D) $x=-\frac{1}{3}$, $y=-\frac{2}{3}$

Concept: Matrix Notation — Two matrices are equal if and only if their corresponding entries are equal.

Step 1: Equate the (1,1) entries.

$3x + 7 = 0 \implies x = -\frac{7}{3}$.

Step 2: Equate the (2,2) entries.

$2 - 3x = 4 \implies -3x = 2 \implies x = -\frac{2}{3}$. …

The idea is simple: for two matrices to be equal, each entry in the first must match the entry in the same position in the second. That gives us a system of equations. But here, the equations conflict — meaning no single pair $(x, y)$ can satisfy all of them at once.

Let’s write the matrices side by side:

Equality means four separate equations:

1. Top-left: $3x + 7 = 0$

   This gives $3x = -7$, so $x = -\frac{7}{3}$.

2. Top-right: $5 = y - 2$

   So $y = 7$.

3. Bottom-left: $y + 1 = 8$

   So $y = 7$ (consistent with the previous).

4. Bottom-right: $2 - 3x = 4$

   This gives $-3x = 2$, so $x = -\frac{2}{3}$.

Now look at the two values for $x$: from equation (1) we got $x = -\frac{7}{3}$, but from equation (4) we got $x = -\frac{2}{3}$. These are different — and that’s a contradiction. …

Method: Checking Consistency of a Matrix Equality for Unknowns

Use this when asked which values make two matrices equal — especially when the answer may be "not possible." A single conflicting entry means no solution exists.

Steps

Step 1: Equate every corresponding entry.

Write one scalar equation per position. Include every cell, because any one of them can override an otherwise tempting answer.

Step 2: Solve for each unknown wherever it appears — and cross-check. …

Common Mistakes

Mistake 1: Stopping after a few entries and picking a value.

Why it's wrong: an unchecked entry (e.g. $2-3x=4$) can contradict an earlier one, making equality impossible. Correct approach: solve every entry and compare before choosing.

Mistake 2: Ignoring that one variable must satisfy all its occurrences.

Why it's wrong: $x$ appearing in two entries must give the same value in both; two different values means no solution. Correct approach: cross-check each shared unknown across all its entries. …