Determinants

We will explore cofactor expansions through some examples.

1. Using a cofactor expansion, show that the determinant of the following matrix . Remember that you can choose any row or column to create the expansion, but the choice of a particular row or column may simplify the computation.

2. Use a cofactor expansion to find the determinant of . Explain how the cofactor expansion technique shows that the de…

Consider the matrices .

1. Find the determinants of $A$ and $B$ using row operations.

2. Now find the determinants of $A$ and $B$ using cofactor expansions to verify your results

In the next chapter, we will say that matrices $A$ and $B$ are similar if there is a matrix $P$ such that $A= PBP^{-1}$.

1. Suppose that $A$ and $B$ are matrices and that there is a matrix $P$ such that $A=PBP^{-1}$. Explain why $\det(A) = \det(B)$.

2. Suppose that $A$ is a $3\times3$ matrix and that there is a matrix $P$ such that . Find $\det(A)$.

Consider the matrix where $k$ is a parameter.

1. Find an expression for $\det(A)$ in terms of the parameter $k$.

2. Use your expression for $\det(A)$ to determine the values of $k$ for which the vectors are linearly independent.

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

1. If we have a square matrix $A$ and multiply the first row by $5$ and add it to the third row to obtain $A'$, then $\det(A') = 5\det(A)$.

2. If we interchange two rows of a matrix, then the determinant is unchanged.

3. If we scale a row of the matrix $A$ by $17$ to obtain $A'$, then $\det(A') = 17\det(A)$.

4. If $A$ and $A'$ are row equivalent and $\det(A') = 0$, then $\det(A) = 0$ also.

5. If $A$ is row equivalent…

Suppose that $A$ and $B$ are $5\times5$ matrices such that $\det(A) = -2$ and $\det(B) = 5$. Find the following determinants:

1. $\det(2A)$.

2. $\det(A^3)$.

3. $\det(AB)$.

4. $\det(-A)$.

5. $\det(AB^{-1})$.

Suppose that $A$ and $B$ are $n\times n$ matrices.

1. If $A$ and $B$ are both invertible, use determinants to explain why $AB$ is invertible.

2. If $AB$ is invertible, use determinants to explain why both $A$ and $B$ are invertible.

Provide a justification for your responses to the following questions.

1. If every entry in one row of a matrix is zero, what can you say about the determinant?

2. If two rows of a square matrix are identical, what can you say about the determinant?

3. If two columns of a square matrix are identical, what can you say about the determinant?

4. If one column of a matrix is a linear combination of the others, what c…

Consider the matrix .

1. Assuming that $\det(A) = 0$, rewrite the equation in terms of $x$, $y$, and $z$.

2. Explain why $\vvec_1$ and $\vvec_2$, the first two columns of $A$, satisfy the equation you found in the previous part.

3. Explain why the solution space of this equation is the plane spanned by $\vvec_1$ and $\vvec_2$.

In this section, we studied the effect of row operations on the matrix $A$. In this exercise, we will study the effect of analogous column operations.

Suppose that $A$ is the $3\times3$ matrix $A= \left[\begin{array}{rrr} \vvec_1 & \vvec_2 & \vvec_3 \end{array}\right]$. Also consider elementary matrices .

1. Explain why the matrix $AR$ is obtained from $A$ by replacing the first column $\vvec_1$ by $\vvec_1 - 3\vvec_3$. We call this a column replacement operation. Explain why column replacement operations do not change…

Consider the matrices . Use row operations to find the determinants of these matrices.

Consider the matrices

1. Use row (and/or column) operations to find the determinants of these matrices.

2. Write the $6\times6$ and $7\times7$ matrices that follow in this pattern and state their determinants based on what you have seen.