Worksheet on Matrix | Solving Matrix Equations Worksheet | Answers - Free Printable

We are given a matrix equation:

$$
\begin{bmatrix}
2 & 3 \\
-4 & 0
\end{bmatrix}
\begin{bmatrix}
x & 1 \\
y & 1
\end{bmatrix}
=
\begin{bmatrix}
4 & z \\
-3 & -4
\end{bmatrix}
$$

We need to find the values of $ x $, $ y $, and $ z $.

---

Step 1: Understand the Matrix Multiplication

We are multiplying two $ 2 \times 2 $ matrices on the left-hand side (LHS), and the result is another $ 2 \times 2 $ matrix on the right-hand side (RHS).

Let’s denote:

$$
A = \begin{bmatrix} 2 & 3 \\ -4 & 0 \end{bmatrix}, \quad
B = \begin{bmatrix} x & 1 \\ y & 1 \end{bmatrix}, \quad
C = \begin{bmatrix} 4 & z \\ -3 & -4 \end{bmatrix}
$$

So, $ A \cdot B = C $

We will compute $ A \cdot B $ and equate it to $ C $, then solve for $ x, y, z $.

---

Step 2: Perform Matrix Multiplication

Recall that to multiply two matrices, we take the dot product of rows of the first matrix with columns of the second.

#### Element (1,1) of result:
$$
(2)(x) + (3)(y) = 2x + 3y
$$
This equals the (1,1) element of RHS: $ 4 $

So:
$$
2x + 3y = 4 \quad \text{(Equation 1)}
$$

#### Element (1,2) of result:
$$
(2)(1) + (3)(1) = 2 + 3 = 5
$$
This equals the (1,2) element of RHS: $ z $

So:
$$
z = 5 \quad \text{(Equation 2)}
$$

#### Element (2,1) of result:
$$
(-4)(x) + (0)(y) = -4x
$$
This equals the (2,1) element of RHS: $ -3 $

So:
$$
-4x = -3 \Rightarrow x = \frac{3}{4} \quad \text{(Equation 3)}
$$

#### Element (2,2) of result:
$$
(-4)(1) + (0)(1) = -4 + 0 = -4
$$
This matches the (2,2) element of RHS: $ -4 $ → ✔️ (consistent)

---

Step 3: Solve for $ y $

From Equation 1:
$$
2x + 3y = 4
$$
Substitute $ x = \frac{3}{4} $:
$$
2\left(\frac{3}{4}\right) + 3y = 4 \\
\frac{6}{4} + 3y = 4 \\
\frac{3}{2} + 3y = 4 \\
3y = 4 - \frac{3}{2} = \frac{8}{2} - \frac{3}{2} = \frac{5}{2} \\
y = \frac{5}{2} \div 3 = \frac{5}{6}
$$

---

Final Answers:

$$
x = \frac{3}{4}, \quad y = \frac{5}{6}, \quad z = 5
$$

---

✔ Verification:

Let’s plug back into the original multiplication:

Left-hand side:
$$
\begin{bmatrix}
2 & 3 \\
-4 & 0
\end{bmatrix}
\begin{bmatrix}
\frac{3}{4} & 1 \\
\frac{5}{6} & 1
\end{bmatrix}
$$

Compute each element:

- (1,1): $ 2 \cdot \frac{3}{4} + 3 \cdot \frac{5}{6} = \frac{6}{4} + \frac{15}{6} = \frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4 $ ✔️
- (1,2): $ 2 \cdot 1 + 3 \cdot 1 = 2 + 3 = 5 $ → $ z = 5 $ ✔️
- (2,1): $ -4 \cdot \frac{3}{4} + 0 \cdot \frac{5}{6} = -3 $ ✔️
- (2,2): $ -4 \cdot 1 + 0 \cdot 1 = -4 $ ✔️

All match!

---

✔ Final Answer:

$$
\boxed{x = \frac{3}{4}, \quad y = \frac{5}{6}, \quad z = 5}
$$