To solve a system of linear equations with matrix equations, we will first write our equations in standard form. Then we will set up three matrices. Once for the coefficients (A), another for the variables (X), and a final matrix (B) for the constants. We can then set up a matrix equation: AX = B. This equation can be solved as: AA(-1)X = A(-1)B
IX = A(-1)B
X = A(-1)B
So we find that X, is just the inverse of matrix A or the coefficient matrix multiplied by matrix B, the constant matrix.

Test Objectives
• Demonstrate the ability to find the inverse of a matrix
• Demonstrate the ability to multiply matrices
• Demonstrate the ability to solve a linear system using matrix equations
Solving Matrix Equations Practice Test:

#1:

Instructions: solving each system.

$$a)\hspace{.2em}$$ $$2x - y=8$$ $$3x - 5y=-2$$

$$b)\hspace{.2em}$$ $$x + 4y=15$$ $$-5x + 6y=3$$

#2:

Instructions: solving each system.

$$a)\hspace{.2em}$$ $$-5x + 3y=11$$ $$x - y=-3$$

$$b)\hspace{.2em}$$ $$-4x + 4y + z=-11$$ $$-6x - 6y - 6z=12$$ $$5y + 5z=-10$$

#3:

Instructions: solving each system.

$$a)\hspace{.2em}$$ $$4x - 3y + 3z=14$$ $$-x - 4y + 3z=24$$ $$-4x - y - 4z=5$$

$$b)\hspace{.2em}$$ $$-2x - 5y + 2z=9$$ $$-x + 2y + 5z=-13$$ $$-5x + 3y - 2z=-17$$

#4:

Instructions: solving each system.

$$a)\hspace{.2em}$$ $$4x + y - 3z=-13$$ $$x + 4y - z=8$$ $$4x - y - 5z=-19$$

Instructions: find X.

$$b)\hspace{.2em}AX=B$$ $$A=\left[ \begin{array}{cc}1 & -2\\ -4 & 3\end{array}\right]$$ $$X=\left[ \begin{array}{c}x \\ y\end{array}\right]$$ $$B=\left[ \begin{array}{c}11 \\ -19\end{array}\right]$$

#5:

Instructions: find X.

$$a)\hspace{.2em}AX=B$$ $$A=\left[ \begin{array}{cc}-5 & 3\\ -5 & 4\end{array}\right]$$ $$X=\left[ \begin{array}{c}x \\ y\end{array}\right]$$ $$B=\left[ \begin{array}{c}-24 \\ -22\end{array}\right]$$

$$b)\hspace{.2em}AX=B$$ $$A=\left[ \begin{array}{ccc}3 & 2 & 1\\ -1 & -3 & -1 \\1 & -2 & -2\end{array}\right]$$ $$X=\left[ \begin{array}{c}x \\ y \\z \end{array}\right]$$ $$B=\left[ \begin{array}{c}-7 \\ 5 \\-4\end{array}\right]$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}(6,4)$$

$$b)\hspace{.2em}(3,3)$$

#2:

Solutions:

$$a)\hspace{.2em}(-1,2)$$

$$b)\hspace{.2em}(0,-3,1)$$

#3:

Solutions:

$$a)\hspace{.2em}(-1,-5,1)$$

$$b)\hspace{.2em}(2,-3,-1)$$

#4:

Solutions:

$$a)\hspace{.2em}(-4,3,0)$$

$$b)\hspace{.2em}X=\left[ \begin{array}{c}1 \\ -5\end{array}\right]$$

#5:

Solutions:

$$a)\hspace{.2em}X=\left[ \begin{array}{c}6 \\ 2\end{array}\right]$$

$$b)\hspace{.2em}X=\left[ \begin{array}{c}-2 \\ -2 \\ 3\end{array}\right]$$