Q.Find the matrix $X$ so that $X\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$

Idea: $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ is $2\times3$ (not invertible), so let $X=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and match entries.

Step 1 — write $XA$.

Step 2 — row 1. $a+4b=-7$ and $2a+5b=-8$. Subtracting twice the first from the second: $-3b = 6 \Rightarrow b=-2$, then $a = -7-4(-2) = 1$. (Check: $3a+6b = 3-12 = -9$ ✓.) …

We need $X$ with $X\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$. Since $A$ is $2\times3$, it has no ordinary inverse, so we can't just multiply by $A^{-1}$. Instead we fix the shape of $X$ and solve for its entries.

Step 1 — shape of $X$

For $XA$ to be defined and to come out $2\times3$ (to match the right side), $X$ must be $2\times2$. Write

Step 2 — form $XA$ and equate

The top row involves only $(a,b)$ and the bottom row only $(c,d)$, so we get two small systems.

Step 3 — solve for $a,b$

$a+4b = -7, \qquad 2a+5b = -8.$ …

Method: Finding an unknown matrix $X$ by comparing entries

Use this when you must find a matrix $X$ from $XA = B$ (or $AX = B$) and $A$ is not square, so no inverse exists.

Steps

Step 1: Deduce the order of $X$ from compatibility.

If $XA$ must equal $B$, match shapes: for $X\,(A_{2\times3}) = B_{2\times3}$, $X$ must be $2\times2$. Write $X$ with unknown entries.

Step 2: Form the product symbolically and equate entry-by-entry. …

Common Mistakes

Mistake 1: Trying to write $X = B A^{-1}$ when $A$ is not square.

Why it's wrong: only square matrices can have an inverse; a $2\times3$ matrix has none. Correct approach: set up $X$ with unknown entries and compare, don't invert.

Mistake 2: Getting the order of $X$ wrong.

Why it's wrong: an incorrect shape makes $XA$ either undefined or the wrong size. Correct approach: fix the rows of $X$ from the rows of $B$ and the columns of $X$ from the rows of $A$.

Mistake 3: Ignoring the extra column equation. …