Q.Find the values of $x, y$ and $z$ from the following equations:

Concept: Matrix Equality — two matrices are equal if and only if their corresponding entries are equal.

(i)

Equate entries:

$4 = y$, $3 = z$, $x = 1$, $5 = 5$ (consistent).

Thus $x = 1$, $y = 4$, $z = 3$.

(ii)

From equality:

$x + y = 6$ …(1)

$2 = 2$ (consistent)

$5 + z = 5 \implies z = 0$

$xy = 8$ …(2)

From (1) and (2): $x$ and $y$ are roots of $t^2 - 6t + 8 = 0 \implies (t-2)(t-4)=0$.

So $(x, y) = (2, 4)$ or $(4, 2)$; $z = 0$.

(iii)

Equate rows:

$x + y + z = 9$ …(1)

$x + z = 5$ …(2)

$y + z = 7$ …(3) …

The Core Idea: Matrix Equality

Two matrices are equal only when they have the same dimensions and every entry in the same position is identical. This turns a matrix equation into a system of ordinary algebraic equations — one equation per matching position.

Let’s apply this to each part.

(i) $\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$

Both are $2 \times 2$ matrices. Match entries position by position:

1. Top-left: $4 = y$ → $y = 4$.
2. Top-right: $3 = z$ → $z = 3$.
3. Bottom-left: $x = 1$ → $x = 1$.
4. Bottom-right: $5 = 5$ — already consistent, no new info.

That’s it. Three unknowns, three equations, all independent. …

Method: Solving for Unknowns Using Matrix Equality

Use this whenever two matrices are set equal and you must find unknowns. Matrix equality converts one matrix equation into a system of scalar equations — one per position.

Steps

Step 1: Confirm equal orders, then equate corresponding entries.

Two matrices are equal iff they have the same order and every corresponding entry matches. Write one scalar equation for each position $(i,j)$.

Step 2: Solve the resulting system, simplest equations first. …

Common Mistakes

Mistake 1: Ignoring a sum-and-product pair's second solution.

Why it's wrong: $x+y=6,\ xy=8$ gives $t^2-6t+8=0$, so $(x,y)=(2,4)$ or $(4,2)$ — both valid. Correct approach: solve via the quadratic and report both orderings unless the problem fixes an order.

Mistake 2: Skipping a "trivial" entry equation.

Why it's wrong: entries like $5+z=5$ carry real information ($z=0$); missing them leaves the system underdetermined. Correct approach: equate every position, then solve. …