Invertibility

This activity demonstrates a procedure for finding the inverse of a matrix $A$.

1. Suppose that $A = \begin{bmatrix} 3 & -2 \\ 1 & -1 \\ \end{bmatrix}$. To find an inverse $B$, we write its columns as $B = \begin{bmatrix}\bvec_1 & \bvec_2 \end{bmatrix}$ and require that In other words, we can find the columns of $B$ by solving the equations Solve these equations to find $\bvec_1$ and $\bvec_2$. Then write the matrix $B$ and verify that $AB=I$. This is enough for us to conclude that $B$ is the inverse of $A$.

2. Find the product $BA$ and expl…

This activity explores how the row operations of scaling, interchange, and replacement can be performed using matrix multiplication.

As an example, we consider the matrix and apply a replacement operation that multiplies the first row by $-2$ and adds it to the second row. Rather than performing this operation in the usual way, we construct a new matrix by applying the desired replacement operation t…

Consider the matrix .

1. Explain why $A$ has an inverse.

2. Find the inverse of $A$ by augmenting by the identity $I$ to form $\left[\begin{array}{r|r}A & I \end{array}\right]$.

3. Use your inverse to solve the equation $A\xvec = \fourvec{3}{2}{-3}{-1}$.

In this exercise, we will consider $2\times 2$ matrices as defining matrix transformations.

1. Write the matrix $A$ that performs a $45^\circ$ rotation. What geometric operation undoes this rotation? Find the matrix that perform this operation and verify that it is $A^{-1}$.

2. Write the matrix $A$ that performs a $180^\circ$ rotation. Verify that $A^2 = I$ so that $A^{-1} = A$, and explain geometrically why this is the case.

3. Find three more matrices $A$ that satisf…

Inverses for certain types of matrices can be found in a relatively straightforward fashion.

1. The matrix $D=\begin{bmatrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -4 \\ \end{bmatrix}$ is called diagonal since the only nonzero entries are on the diagonal of the matrix.

   1. Find $D^{-1}$ by augmenting $D$ by the identity and finding its reduced row echelon form.

   2. Under what conditions is a diagonal matrix invertible?

   3. Explain why the inverse of a diagonal matrix is also diagonal and explain the…

Our definition of an invertible matrix requires that $A$ be a square $n\times n$ matrix. Let's examine what happens when $A$ is not square. For instance, suppose that .

1. Verify that $BA = I_2$. In this case, we say that $B$ is a left inverse of $A$.

2. If $A$ has a left inverse $B$, we can still use it to find solutions to linear equations. If we know there is a solution to the equation $A\xvec = \bvec$, we can multiply both sides of the equation b…

If a matrix $A$ is invertible, there is a sequence of row operations that transforms $A$ into the identity matrix $I$. We have seen that every row operation can be performed by matrix multiplication. If the $j^{th}$ step in the Gaussian elimination process is performed by multiplying by $E_j$, then we have , which means that . For each of the following matrices, find a sequence of row operations that transforms th…

Suppose that $A$ is an $n\times n$ matrix.

1. Suppose that $A^2 = AA$ is invertible with inverse $B$. This means that $A^2B = AAB = I$. Explain why $A$ must be invertible with inverse $AB$.

2. Suppose that $A^{100}$ is invertible with inverse $B$. Explain why $A$ is invertible. What is $A^{-1}$ in terms of $A$ and $B$?

Determine whether the following statements are true or false and explain your reasoning.

1. If $A$ is invertible, then the columns of $A$ are linearly independent.

2. If $A$ is a square matrix whose diagonal entries are all nonzero, then $A$ is invertible.

3. If $A$ is an invertible $n\times n$ matrix, then span of the columns of $A$ is $\real^n$.

4. If $A$ is invertible, then there is a nonzero solution to the homogeneous equation $A\xvec = \zerovec$.

5. If $A$ is an…

Provide a justification for your response to the following questions.

1. Suppose that $A$ is a square matrix with two identical columns. Can $A$ be invertible?

2. Suppose that $A$ is a square matrix with two identical rows. Can $A$ be invertible?

3. Suppose that $A$ is an invertible matrix and that $AB = AC$. Can you conclude that $B = C$?

4. Suppose that $A$ is an invertible $n\times n$ matrix. What can you say about the span of the columns of $A^{-1}$?

5. S…

Suppose that we start with the $3\times3$ matrix $A$, perform the following sequence of row operations:

1. Multiply row 1 by -2 and add to row 2.

2. Multiply row 1 by 4 and add to row 3.

3. Scale row 2 by $1/2$.

4. Multiply row 2 by -1 and add to row 3,

and arrive at the upper triangular matrix

1. Write the matrices $E_1$, $E_2$, $E_3$, and $E_4$ that perform the four row operations.

2. Find the matrix $E = E_4E_3E_2E_1$.

3. We then have $E_4E_3E_2E_1 A = EA = U$. Now that we have the matrix $E$,…

We say that two square matrices $A$ and $B$ are similar if there is an invertible matrix $P$ such that $B = PAP^{-1}$.

1. If $A$ and $B$ are similar, explain why $A^2$ and $B^2$ are similar as well. In particular, if $B = PAP^{-1}$, explain why $B^2 = PA^2P^{-1}$.

2. If $A$ and $B$ are similar and $A$ is invertible, explain why $B$ is also invertible.

3. If $A$ and $B$ are similar and both are invertible, explain why $A^{-1}$ and $B^{-1}$ are similar.

4. If $A$ is similar to $B$ and $B$ is similar to $C$,…

Suppose that $A$ and $B$ are two $n\times n$ matrices and that $AB$ is invertible. We would like to explain why both $A$ and $B$ are invertible.

1. We first explain why $B$ is invertible.

   1. Since $AB$ is invertible, explain why any solution to the homogeneous equation $AB\xvec = \zerovec$ is $\xvec=\zerovec$.

   2. Use this fact to explain why any solution to $B\xvec = \zerovec$ must be $\xvec=\zerovec$.

   3. Explain why $B$ must be invertible.

2. Now we explain why $A$ is invertible.

   1. Since $AB$ is invertible, explain w…