7. Systems of Equations & Matrices

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases}x - y + z = 8 \\2y - z = -7 \\2x + 3y = 1\end{cases}\\\text{The inverse of }\begin{bmatrix}1 & -1 & 1 \\0 & 2 & -1 \\2 & 3 & 0\end{bmatrix}\text{ is }\begin{bmatrix}3 & 3 & -1 \\-2 & -2 & 1 \\-4 & -5 & 2\end{bmatrix}\text{.}$

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases}2x + 6y + 6z = 8 \\2x + 7y + 6z = 10 \\2x + 7y + 7z = 9\end{cases}\\\text{The inverse of }\begin{bmatrix}2 & 6 & 6 \\2 & 7 & 6 \\2 & 7 & 7\end{bmatrix}\text{ is }\begin{bmatrix}\frac{7}{2} & 0 & -3 \\-1 & 0 & 0 \\0 & -1 & 1\end{bmatrix}\text{.}$

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = \begin{bmatrix}4 & -3 \\-5 & 4\end{bmatrix}, B = \begin{bmatrix}4 & 3 \\5 & 4\end{bmatrix}$

Write each matrix equation as a system of linear equations without matrices.

$\begin{bmatrix}2 & 0 & -1 \\0 & 3 & 0 \\1 & 1 & 0\end{bmatrix}\begin{bmatrix}x \\y \\z\end{bmatrix}=\begin{bmatrix}6 \\9 \\5\end{bmatrix}$

Write each matrix equation as a system of linear equations without matrices.

$\begin{bmatrix}4 & -7 \\2 & -3\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}=\begin{bmatrix}-3 \\1\end{bmatrix}$

Use the fact that if , then to find the inverse of each matrix, if possible. Check that $A{A}^{-1}=I_2$ and $A^{-1}A=I_2$.

$A = \begin{bmatrix}10 & -2 \\-5 & 1\end{bmatrix}$

Use the fact that if , then to find the inverse of each matrix, if possible. Check that $A{A}^{-1}=I_2$ and $A^{-1}A=I_2$.

$A = \begin{bmatrix}3 & -1 \\-4 & 2\end{bmatrix}$

Use the fact that if , then to find the inverse of each matrix, if possible. Check that $A{A}^{-1}=I_2$ and $A^{-1}A=I_2$.

In Exercises 43–44, (a) Write each linear system as a matrix equation in the form AX = B (b) Solve the system using the inverse that is given for the coefficient matrix.

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

Answer each question. What is the product of and I₂ (in either order)?

Answer each question. What is the product ?