How to Solve Matrix Equations

What is a Matrix Equation?

A matrix equation has the form $$AX \ = \ B$$, where $$A$$ is the coefficient matrix, X is the column matrix of variables, and $$B$$ is the column matrix of constants on the right side of the equations in a system.

Solving Matrix Equation?

Let's figure out what $$X$$ is in the equation $$AX \ = \ B$$. We did this by multiplying each side of the equation by ($$A^{-1}$$). (the inverse of $$A$$)

$$AX \ = \ B \ ⇒ \ A^{-1} \ (AX) \ = \ A^{-1} \ B$$

We know that $$A^{-1} A$$ equals $$I$$, where $$I$$ is the same-order identity matrix as $$A$$. So,

$$IX \ = \ A^{-1} \ B$$

Also we know that: $$IX \ = \ X$$. So,

$$X \ = \ A^{-1} \ B$$

The answer to the matrix equation is found here. This is also called the "inverse matrix equation," and the way to solve a set of equations by using the above formula is called the "inverse matrix method." So, here are the steps to use matrices to solve a set of equations:

• Put the system into the form of the matrix equation $$AX \ = \ B$$.
• Find the Inverse of $$A$$.
• To get the answer, multiply it by the constant matrix $$B$$. i.e., $$X \ = \ A^{-1}B$$.

Exercises for Solving Matrix Equations

1) Find the answer: $$X \ + \ \begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} \ = \ \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix}$$

2) Find the answer: $$X \ - \ \begin{bmatrix} 11 & 3 \\\ -5 & 8 \end{bmatrix} \ = \ \begin{bmatrix} 13 & 21 \\\ -7 & 6 \end{bmatrix}$$

3) Find the answer: $$2Y \ + \ \begin{bmatrix} 9 & -23 \\\ -15 & -7 \end{bmatrix} \ = \ \begin{bmatrix} -8 & -27 \\\ 13 & 3 \end{bmatrix}$$

4) Find the answer: $$2Y \ - \ \begin{bmatrix} -15 & 21 \\\ 19 & -17 \end{bmatrix} \ = \ \begin{bmatrix} 9 & 16 \\\ -12 & 7 \end{bmatrix}$$

5) Find the answer: $$2X \ + \ \begin{bmatrix} -7 & 11 \\\ -7 & -7 \end{bmatrix} \ = \ \begin{bmatrix} 15 & -29 \\\ 17 & -11 \end{bmatrix}$$

6) Find the answer: $$2X \ + \ \begin{bmatrix} 19 & -20 & 8 \end{bmatrix} \ = \ \begin{bmatrix} 25 & -31 & -12 \end{bmatrix}$$

7) Find the answer: $$X \times \begin{bmatrix} 2 & -4 \\\ 1 & 3 \end{bmatrix} \ = \ \begin{bmatrix} 3 & -1 \\\ 2 & -3 \end{bmatrix}$$

8) Find the answer: $$X \times \begin{bmatrix} 5 & 1 \\\ 2 & -3 \end{bmatrix} \ = \ \begin{bmatrix} 8 & 6 \\\ 9 & 12 \end{bmatrix}$$

9) Find the answer: $$-3X \ = \ \begin{bmatrix} -18 & 33 & -69 \end{bmatrix}$$

10) Find the answer: $$7 \ Y \ = \ \begin{bmatrix} 70 & 42 & -56 \end{bmatrix}$$

1) Find the answer: $$X \ + \ \begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} \ = \ \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix}$$

$$\color{red}{X \ = \ \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix} \ - \ \begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} \ = \ \begin{bmatrix} -6 & 3 \\\ -3 & -12 \end{bmatrix}}$$

2) Find the answer: $$X \ - \ \begin{bmatrix} 11 & 3 \\\ -5 & 8 \end{bmatrix} \ = \ \begin{bmatrix} 13 & 21 \\\ -7 & 6 \end{bmatrix}$$

$$\color{red}{X \ = \ \begin{bmatrix} 11 & 3 \\\ -5 & 8 \end{bmatrix} \ + \ \begin{bmatrix} 13 & 21 \\\ -7 & 6 \end{bmatrix} \ = \ \begin{bmatrix} 24 & 24 \\\ -12 & 14 \end{bmatrix}}$$

3) Find the answer: $$2Y \ + \ \begin{bmatrix} 9 & -23 \\\ -15 & -7 \end{bmatrix} \ = \ \begin{bmatrix} -8 & -27 \\\ 13 & 3 \end{bmatrix}$$

$$\color{red}{2Y \ = \ \begin{bmatrix} -8 & -27 \\\ 13 & 3 \end{bmatrix} \ - \ \begin{bmatrix} 9 & -23 \\\ -15 & -7 \end{bmatrix} \ = \ \begin{bmatrix} -17 & -4 \\\ 28 & 10 \end{bmatrix} \ ⇒ \ Y \ = \ \begin{bmatrix} -8.5 & -2 \\\ 14 & 5 \end{bmatrix}}$$

4) Find the answer: $$2Y \ - \ \begin{bmatrix} -15 & 21 \\\ 19 & -17 \end{bmatrix} \ = \ \begin{bmatrix} 9 & 16 \\\ -12 & 7 \end{bmatrix}$$

$$\color{red}{2Y \ = \ \begin{bmatrix} -15 & 21 \\\ 19 & -17 \end{bmatrix} \ + \ \begin{bmatrix} 9 & 16 \\\ -12 & 7 \end{bmatrix} \ = \ \begin{bmatrix} -6 & 37 \\\ 7 & -10 \end{bmatrix} \ ⇒ \ Y \ = \ \begin{bmatrix} -3 & 18.5 \\\ 3.5 & -5 \end{bmatrix}}$$

5) Find the answer: $$2X \ + \ \begin{bmatrix} -7 & 11 \\\ -7 & -7 \end{bmatrix} \ = \ \begin{bmatrix} 15 & -29 \\\ 17 & -11 \end{bmatrix}$$

$$\color{red}{2X \ = \ \begin{bmatrix} 15 & -29 \\\ 17 & -11 \end{bmatrix} \ - \ \begin{bmatrix} -7 & 11 \\\ -7 & -7 \end{bmatrix} \ = \ \begin{bmatrix} 22 & -40 \\\ 24 & -4 \end{bmatrix} \ ⇒ \ X \ = \ \begin{bmatrix} 11 & -20 \\\ 12 & -2 \end{bmatrix}}$$

6) Find the answer: $$2X \ + \ \begin{bmatrix} 19 & -20 & 8 \end{bmatrix} \ = \ \begin{bmatrix} 25 & -31 & -12 \end{bmatrix}$$

$$\color{red}{2X \ = \ \begin{bmatrix} 25 & -31 & -12 \end{bmatrix} \ - \ \begin{bmatrix} 19 & -20 & 8 \end{bmatrix} \ = \ \begin{bmatrix} 6 & -11 & -20 \end{bmatrix}}$$ $$\color{red}{ ⇒ \ X \ = \ \begin{bmatrix} 3 & -5.5 & -10 \end{bmatrix}}$$

7) Find the answer: $$X \times \begin{bmatrix} 2 & -4 \\\ 1 & 3 \end{bmatrix} \ = \ \begin{bmatrix} 3 & -1 \\\ 2 & -3 \end{bmatrix}$$

$$\color{red}{X \times \begin{bmatrix} 2 & -4 \\\ 1 & 3 \end{bmatrix} \ = \ \begin{bmatrix} 3 & -1 \\\ 2 & -3 \end{bmatrix} \ ⇒ \ X \ = \ \begin{bmatrix} 3 & -1 \\\ 2 & -3 \end{bmatrix} \times \begin{bmatrix} 2 & -4 \\\ 1 & 3 \end{bmatrix}^{-1}}$$ $$\color{red}{ \ = \ \begin{bmatrix} 3 & -1 \\\ 2 & -3 \end{bmatrix} \times \begin{bmatrix} \frac{3}{10} & \frac{2}{5} \\\ -\frac{1}{10} & \frac{1}{5} \end{bmatrix}= \ \begin{bmatrix} \frac{17}{10} & -\frac{3}{2} \\\ \frac{1}{10} & -\frac{1}{2} \end{bmatrix}}$$

8) Find the answer: $$X \times \begin{bmatrix} 5 & 1 \\\ 2 & -3 \end{bmatrix} \ = \ \begin{bmatrix} 8 & 6 \\\ 9 & 12 \end{bmatrix}$$

$$\color{red}{X \times \begin{bmatrix} 5 & 1 \\\ 2 & -3 \end{bmatrix} \ = \ \begin{bmatrix} 8 & 6 \\\ 9 & 12 \end{bmatrix} \ ⇒ \ X \ = \ \begin{bmatrix} 8 & 6 \\\ 9 & 12 \end{bmatrix} \times \begin{bmatrix} 5 & 1 \\\ 2 & -3 \end{bmatrix}^{-1}}$$ $$\color{red}{ \ = \ \begin{bmatrix} 8 & 6 \\\ 9 & 12 \end{bmatrix} \times \begin{bmatrix} \frac{3}{17} & \frac{1}{17} \\\ \frac{2}{17} & -\frac{5}{17} \end{bmatrix}= \ \begin{bmatrix} \frac{33}{17} & \frac{30}{17} \\\ -\frac{29}{17} & -\frac{48}{17} \end{bmatrix}}$$

9) Find the answer: $$-3X \ = \ \begin{bmatrix} -18 & 33 & -69 \end{bmatrix}$$

$$\color{red}{-3X \ = \ \begin{bmatrix} -18 & 33 & -69 \end{bmatrix} \ ⇒ \ X \ = \ \begin{bmatrix} 6 & -11 & 23 \end{bmatrix}}$$

10) Find the answer: $$7 \ Y \ = \ \begin{bmatrix} 70 & 42 & -56 \end{bmatrix}$$

$$\color{red}{7 \ Y \ = \ \begin{bmatrix} 70 & 42 & -56 \end{bmatrix} \ ⇒ \ Y \ = \ \begin{bmatrix} 10 & 6 & -8 \end{bmatrix}}$$

Matrix Equations Practice Quiz

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