Matrix Equations worksheet with six problems for solving variables in matrix equations.

Let's solve each matrix equation step by step. We are solving for the unknown matrix variable (K, Q, P, G, S, H) in each problem.

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1)

$$
\begin{bmatrix} 16 & -7 & 17 \end{bmatrix} + 3K = \begin{bmatrix} -2 & -1 & -4 \end{bmatrix}
$$

Step 1: Subtract the known matrix from both sides:
$$
3K = \begin{bmatrix} -2 & -1 & -4 \end{bmatrix} - \begin{bmatrix} 16 & -7 & 17 \end{bmatrix}
$$

$$
3K = \begin{bmatrix} -2 - 16 & -1 - (-7) & -4 - 17 \end{bmatrix} = \begin{bmatrix} -18 & 6 & -21 \end{bmatrix}
$$

Step 2: Divide by 3:
$$
K = \frac{1}{3} \begin{bmatrix} -18 & 6 & -21 \end{bmatrix} = \begin{bmatrix} -6 & 2 & -7 \end{bmatrix}
$$

✔ Answer: $ K = \begin{bmatrix} -6 & 2 & -7 \end{bmatrix} $

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2)

$$
\begin{bmatrix} 1 \\ 14 \end{bmatrix} + 4Q = \begin{bmatrix} 5 \\ -6 \end{bmatrix}
$$

Step 1: Subtract:
$$
4Q = \begin{bmatrix} 5 \\ -6 \end{bmatrix} - \begin{bmatrix} 1 \\ 14 \end{bmatrix} = \begin{bmatrix} 4 \\ -20 \end{bmatrix}
$$

Step 2: Divide by 4:
$$
Q = \frac{1}{4} \begin{bmatrix} 4 \\ -20 \end{bmatrix} = \begin{bmatrix} 1 \\ -5 \end{bmatrix}
$$

✔ Answer: $ Q = \begin{bmatrix} 1 \\ -5 \end{bmatrix} $

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3)

$$
\begin{bmatrix} -34 & -2 \\ 28 & -33 \end{bmatrix} - 5P = \begin{bmatrix} 1 & 3 \\ -2 & -3 \end{bmatrix}
$$

Step 1: Move $ 5P $ to the right and others to the left:
$$
-5P = \begin{bmatrix} 1 & 3 \\ -2 & -3 \end{bmatrix} - \begin{bmatrix} -34 & -2 \\ 28 & -33 \end{bmatrix}
$$

$$
-5P = \begin{bmatrix} 1 + 34 & 3 + 2 \\ -2 - 28 & -3 + 33 \end{bmatrix} = \begin{bmatrix} 35 & 5 \\ -30 & 30 \end{bmatrix}
$$

Step 2: Multiply both sides by $-1$:
$$
5P = \begin{bmatrix} -35 & -5 \\ 30 & -30 \end{bmatrix}
$$

Step 3: Divide by 5:
$$
P = \frac{1}{5} \begin{bmatrix} -35 & -5 \\ 30 & -30 \end{bmatrix} = \begin{bmatrix} -7 & -1 \\ 6 & -6 \end{bmatrix}
$$

✔ Answer: $ P = \begin{bmatrix} -7 & -1 \\ 6 & -6 \end{bmatrix} $

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4)

$$
\begin{bmatrix} -15 & 3 & -2 \\ -13 & 4 & -2 \end{bmatrix} - 3G = \begin{bmatrix} 6 & -6 & 4 \\ -1 & 5 & -5 \end{bmatrix}
$$

Step 1: Move $ 3G $ to the right:
$$
-3G = \begin{bmatrix} 6 & -6 & 4 \\ -1 & 5 & -5 \end{bmatrix} - \begin{bmatrix} -15 & 3 & -2 \\ -13 & 4 & -2 \end{bmatrix}
$$

$$
-3G = \begin{bmatrix} 6 + 15 & -6 - 3 & 4 + 2 \\ -1 + 13 & 5 - 4 & -5 + 2 \end{bmatrix} = \begin{bmatrix} 21 & -9 & 6 \\ 12 & 1 & -3 \end{bmatrix}
$$

Step 2: Multiply both sides by $-1$:
$$
3G = \begin{bmatrix} -21 & 9 & -6 \\ -12 & -1 & 3 \end{bmatrix}
$$

Step 3: Divide by 3:
$$
G = \frac{1}{3} \begin{bmatrix} -21 & 9 & -6 \\ -12 & -1 & 3 \end{bmatrix} = \begin{bmatrix} -7 & 3 & -2 \\ -4 & -\frac{1}{3} & 1 \end{bmatrix}
$$

✔ Answer: $ G = \begin{bmatrix} -7 & 3 & -2 \\ -4 & -\frac{1}{3} & 1 \end{bmatrix} $

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5)

$$
\begin{bmatrix} 5 \\ -18 \\ -25 \end{bmatrix} + 3S = \begin{bmatrix} 2 \\ -6 \\ -7 \end{bmatrix}
$$

Step 1: Subtract:
$$
3S = \begin{bmatrix} 2 \\ -6 \\ -7 \end{bmatrix} - \begin{bmatrix} 5 \\ -18 \\ -25 \end{bmatrix} = \begin{bmatrix} -3 \\ 12 \\ 18 \end{bmatrix}
$$

Step 2: Divide by 3:
$$
S = \frac{1}{3} \begin{bmatrix} -3 \\ 12 \\ 18 \end{bmatrix} = \begin{bmatrix} -1 \\ 4 \\ 6 \end{bmatrix}
$$

✔ Answer: $ S = \begin{bmatrix} -1 \\ 4 \\ 6 \end{bmatrix} $

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6)

$$
\begin{bmatrix} 13 & -3 \\ 22 & -10 \\ -18 & 24 \end{bmatrix} + 4H = \begin{bmatrix} -3 & -7 \\ -2 & -6 \\ 2 & -4 \end{bmatrix}
$$

Step 1: Subtract:
$$
4H = \begin{bmatrix} -3 & -7 \\ -2 & -6 \\ 2 & -4 \end{bmatrix} - \begin{bmatrix} 13 & -3 \\ 22 & -10 \\ -18 & 24 \end{bmatrix}
$$

$$
4H = \begin{bmatrix} -3 - 13 & -7 + 3 \\ -2 - 22 & -6 + 10 \\ 2 + 18 & -4 - 24 \end{bmatrix} = \begin{bmatrix} -16 & -4 \\ -24 & 4 \\ 20 & -28 \end{bmatrix}
$$

Step 2: Divide by 4:
$$
H = \frac{1}{4} \begin{bmatrix} -16 & -4 \\ -24 & 4 \\ 20 & -28 \end{bmatrix} = \begin{bmatrix} -4 & -1 \\ -6 & 1 \\ 5 & -7 \end{bmatrix}
$$

✔ Answer: $ H = \begin{bmatrix} -4 & -1 \\ -6 & 1 \\ 5 & -7 \end{bmatrix} $

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✔ Final Answers:

1) $ K = \begin{bmatrix} -6 & 2 & -7 \end{bmatrix} $
2) $ Q = \begin{bmatrix} 1 \\ -5 \end{bmatrix} $
3) $ P = \begin{bmatrix} -7 & -1 \\ 6 & -6 \end{bmatrix} $
4) $ G = \begin{bmatrix} -7 & 3 & -2 \\ -4 & -\frac{1}{3} & 1 \end{bmatrix} $
5) $ S = \begin{bmatrix} -1 \\ 4 \\ 6 \end{bmatrix} $
6) $ H = \begin{bmatrix} -4 & -1 \\ -6 & 1 \\ 5 & -7 \end{bmatrix} $

Let me know if you'd like these formatted neatly or printed!