Evaluate each of the following determinants by inspection:

1. $\left|\begin{array}{ccc}0& 0& 3\\ 0& 4& 1\\ 2& 3& 1\end{array}\right|$

2. $\left|\begin{array}{cccc}\hfill 1& \hfill 1& \hfill 1& 3\\ \hfill 0& \hfill 3& \hfill 1& 1\\ \hfill 0& \hfill 0& \hfill 2& 2\\ \hfill -1& \hfill -1& \hfill -1& 2\end{array}\right|$

3. $\left|\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right|$

Let

1. Use the elimination method to evaluate det(A).

2. Use the value of det(A) to evaluate

   $$\left|\begin{array}{rrrr}0& 1& 2& 3\\ -2& -2& 3& 3\\ 1& 2& -2& -3\\ 1& 1& 1& 1\end{array}\right|+\left|\begin{array}{rrrr}0& 1& 2& 3\\ 1& 1& 1& 1\\ -1& -1& 4& 4\\ 2& 3& -1& -2\end{array}\right|$$

For each of the following, compute the determinant and state whether the matrix is singular or nonsingular:

1. $\left[\begin{array}{cc}3& 1\\ 6& 2\end{array}\right]$

2. $\left[\begin{array}{cc}3& 1\\ 4& 2\end{array}\right]$

3. $\left[\begin{array}{ccc}3& 3& 1\\ 0& 1& 2\\ 0& 2& 3\end{array}\right]$

4. $\left[\begin{array}{ccc}2& 1& 1\\ 4& 3& 5\\ 2& 1& 2\end{array}\right]$

5. $\left[\begin{array}{rrr}2& -1& 3\\ -1& 2& -2\\ 1& 4& 0\end{array}\right]$

6. $\left[\begin{array}{cccc}1& \hfill 1& 1& 1\\ 2& \hfill -1& 3& 2\\ 0& \hfill 1& 2& 1\\ 0& \hfill 0& 7& 3\end{array}\right]$

Find all possible choices of c that would make the following matrix singular:

Let A be an $n\times n$ matrix and $\alpha$ a scalar. Show that

Let A be a nonsingular matrix. Show that

Let A and B be $3\times 3$ matrices with $\text{det}(A)=4$ and $\text{det}(B)=5$. Find the value of

1. det(AB)

2. det(3A)

3. det(2AB)

4. $\text{det}(A^{-1}B)$

Show that if E is an elementary matrix, then $E^T$ is an elementary matrix of the same type as E.

Let $E_1$, $E_2$, and $E_3$ be $3\times 3$ elementary matrices of types I, II, and III, respectively, and let A be a $3\times 3$ matrix with $\text{det}(A)=6$. Assume, additionally, that $E_2$ was formed from I by multiplying its second row by 3. Find the values of each of the following:

1. $\text{det}(E_{1}A)$

2. $\text{det}(E_{2}A)$

3. $\text{det}(E_{3}A)$

4. $\text{det}({AE}_1)$

5. $\text{det}(E_1^2)$

6. $\text{det}(E_1{E}_2{E}_3)$

Let A and B be row equivalent matrices, and suppose that B can be obtained from A by using only row operations I and III. How do the values of det(A) and det(B) compare? How will the values compare if B can be obtained from A using only row operation III? Explain your answers.

Let A be an $n\times n$ matrix. Is it possible for $A^2+I=O$ in the case where n is odd? Answer the same question in the case where n is even.

Consider the $3\times 3$ Vandermonde matrix

1. Show that $V=(x_2-x_1)(x_3-x_1)(x_3-x_2)$. Hint: Make use of row operation III.

2. What conditions must the scalars $x_1$, $x_2$, and $x_3$ satisfy in order for V to be nonsingular?

Suppose that a $3\times 3$matrix A factors into a product:

Determine the value of det(A).

Let A and B be $n\times n$ matrices. Prove that the product AB is nonsingular if and only if A and B are both nonsingular.

Let A and B be $n\times n$ matrices. Prove that if $AB=I$, then $BA=I$. What is the significance of this result in terms of the definition of a nonsingular matrix?

A matrix A is said to be skew symmetric if $A^T=-A$. For example,

is skew symmetric, since

If A is an $n\times n$ skew-symmetric matrix and n is odd, show that A must be singular.

Let A be a nonsingular $n\times n$ matrix with a nonzero cofactor $A_{\mathit{\text{nn}}}$, and set

Show that if we subtract c from $a_{\mathit{\text{nn}}}$, then the resulting matrix will be singular.

Let A be a $k\times k$ matrix and let B be an $(n-k)\times (n-k)$ matrix. Let

where $I_k$ and $I_{n-k}$ are the $k\times k$ and $(n-k)\times (n-k)$ identity matrices.

1. Show that $\text{det}(E)=\text{det}(B)$.

2. Show that $\text{det}(F)=\text{det}(A)$.

3. Show that $\text{det}(C)=\text{det}(A)\text{ det}(B)$.

Let A and B be $k\times k$ matrices and let

Show that $\text{det}(M)={(-1)}^k\text{ det}(A)\text{ det}(B)$.

Show that evaluating the determinant of an $n\times n$ matrix by cofactors involves $(n\mathrm{!}-1)$ additions and $\sum _{k=1}^{n-1}n!/k!$ multiplications.

Show that the elimination method of computing the value of the determinant of an $n\times n$ matrix involves $\left[n(n-1)(2n-1)\right]/6$ additions and $\left[n(n-1)(n^2+n+3)\right]/3$ multiplications and divisions. Hint:Atthe ith step of the reduction process, it takes $n-i$ divisions to calculate the multiples of the ith row that are to be subtracted from the remaining rows below the pivot. We must then calculate new values for the ${(n-i)}^2$ entries in rows $i+1$ through n and columns $i+1$ through n.