Properties of determinants

Let

An operation is done to get from $A$ to a matrix $B$. In each case, identify which operation was done and explain how it will affect the value of the determinant.

Let $A$ be an $n\times n$-matrix and suppose there are $n-1$ rows such that the remaining row is a linear combinations of these $n-1$ rows. Show $\det(A)=0$.

Let $A$ be an $n\times n$-matrix. Show that if $\det(A)\neq 0$ and $A\mathbf{x}=\mathbf{0}$, then $\mathbf{x}=\mathbf{0}$.

Using only Theorems 7.23 and 7.39, show that $\det(kA)=k^{n}\det(A)$ for an $n\times n$-matrix $A$ and scalar $k$.

Construct two random $2\times 2$-matrices $A$ and $B$ and verify that $\det(A)\det(B)=\det(AB)$.

Is it true that $\det(A+B)=\det(A)+\det(B)$? If this is so, explain why. If it is not so, give a counterexample.

An $n\times n$-matrix is called nilpotent if there exists some positive integer $k$ such that $A^{k}=0$. If $A$ is a nilpotent matrix, what are the possible values of $\det(A)$?

A square matrix is said to be orthogonal if $A^{T}A=I$. Thus the inverse of an orthogonal matrix is its transpose. What are the possible values of $\det(A)$ if $A$ is an orthogonal matrix?

Let $A$ and $B$ be two $n\times n$-matrices. We say that $A$ is similar to $B$, in symbols $A\sim B$, if there exists an invertible matrix $P$ such that $A=P^{-1}BP$. Show that if $A\sim B$, then $\det(A)=\det(B)$.

Find the determinant of

For which values of $a$ and $b$ is this matrix invertible? Hint: after you compute the determinant, you can factor out $(a-1)$ and $(b-1)$ from it.

Assume $A$, $B$, and $C$ are $n\times n$-matrices and $ABC$ is invertible. Use determinants to show that each of $A,B$, and $C$ is invertible.

Suppose $A$ is an upper triangular matrix. Show that $A^{-1}$ exists if and only if all elements of the main diagonal are non-zero. Is it true that $A^{-1}$ will also be upper triangular? Explain. Could the same be concluded for lower triangular matrices?

Specify whether each statement is true or false. If true, provide a proof. If false, provide a counterexample.

1. If $A$ is a $3\times 3$-matrix with determinant zero, then one column must be a multiple of some other column.

2. If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.

3. For two $n\times n$-matrices $A$ and $B$, $\det(A+B)=\det(A)+\det(B)$.

4. For an $n\times n$-matrix $A$, $\det(3A)=3\det(A)$.

5. If $A^{-1}$ exists, then $\det(A^{-1})=\det(A)^{-1}$.

6. If $B$ is…