2.2 Exercises

20 points · 4 extra credit

The Inverse of a Matrix

1. 1. (2 pts) Use the $2\times2$ inverse formula to find $A^{-1}$, where $A = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}$.

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2. 2. (3 pts) For which value of $k$ is $A = \begin{bmatrix} k & 2 \\ 8 & 4 \end{bmatrix}$ not invertible? For every other value of $k$, give $A^{-1}$ in terms of $k$.

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3. 3. (3 pts) Find the inverse of

   $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 3 \\ 3 & 7 & 8 \end{bmatrix}$

   by row reducing the augmented matrix $[\,A \mid I\,]$.

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4. 4. (2 pts) Using $A = \begin{bmatrix} 7 & 3 \\ 2 & 1 \end{bmatrix}$ (whose inverse is $\begin{bmatrix} 1 & -3 \\ -2 & 7 \end{bmatrix}$), solve $A\mathbf{x} = \mathbf{b}$ for $\mathbf{b} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}$ using the inverse.

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5. 5. (3 pts) Let $A$, $B$, $X$ be $n\times n$ matrices with $A$ and $B$ invertible. Solve each equation for $X$:

   (a) $AXB = I$.

   (b) $(X + A)^{-1} = B$.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) If $A$ is invertible, then $A^{\mathsf T}$ is invertible and $(A^{\mathsf T})^{-1} = (A^{-1})^{\mathsf T}$.

   (b) A product of invertible $n\times n$ matrices is invertible.

   (c) If $A$ is an $n\times n$ matrix with $A^2 = I$, then $A = I$ or $A = -I$.

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7. 7. (2 pts) Show that if $A$ is invertible, then for every $\mathbf{b}$ the equation $A\mathbf{x} = \mathbf{b}$ has the unique solution $\mathbf{x} = A^{-1}\mathbf{b}$.

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8. 8. (2 pts) Explain why a square matrix that has a column of all zeros cannot be invertible.

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9. 9. (4 pts ••) Prove the following two facts about invertible matrices.

   (a) If $A$ is invertible, its inverse is unique (there is only one matrix $B$ with $AB = BA = I$).

   (b) If $A$ and $B$ are invertible $n\times n$ matrices, then $AB$ is invertible and $(AB)^{-1} = B^{-1}A^{-1}$.

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