Back to Exercise: Use matrix inverses to solve systems

Exercises: Find Matrix Inverses and Solve Systems

For all inverse computations, verify your result by checking that AA⁻¹ = I. For system solutions, verify by substituting into the original equations.

Grade 11·18 problems·~40 min·Common Core Math - HS Algebra·standard·hsa-rei-c-9

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Fluency Practice

Let $A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$ .

Verify that $AA^{-1} = I$ by computing the product.

$A diagram showing a 2x2 matrix [a,b;c,d] with three labeled steps: Step 1 Swap a and d, Step 2 Negate b and c, Step 3 Divide by det(A) = ad minus bc.$

Use the 2×2 inverse formula to find $A^{-1}$ for $A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$ .

Steps: (1) compute $\det(A)$ , (2) apply the formula $A^{-1} = \frac{1}{\det(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ , (3) verify $AA^{-1} = I$ .

Find $A^{-1}$ for $A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}$ . Show all steps including the determinant calculation.

For which matrix does an inverse NOT exist?

$\begin{bmatrix} 5 & 2 \\ 3 & 1 \end{bmatrix}$ , because $\det = 5 - 6 = -1$

$\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$ , because $\det = 6 - 6 = 0$

$\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix}$ , because it is a diagonal matrix

$\begin{bmatrix} -2 & 4 \\ 1 & -1 \end{bmatrix}$ , because it has negative entries

Use $A^{-1} = \begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}$ to solve the system $3x + y = 11$ and $5x + 2y = 19$ . Show the matrix multiplication $\mathbf{x} = A^{-1}\mathbf{b}$ .

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Exercises: Find Matrix Inverses and Solve Systems

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Fluency Practice