91Ó°ÊÓ

Find the following determinant by expanding along the first column and third row. $$\left|\begin{array}{lll}1 & 2 & 1 \\\1 & 0 & 1 \\\2 & 1 & 1\end{array}\right|$$

For the following matrices, determine if they are invertible. If so, use the formula for the inverse in terms of the cofactor matrix to find each inverse. If the inverse does not exist, explain why. (a) \(\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]\) $$\begin{array}{l}\text { (b) }\left[\begin{array}{lll}1 & 2 & 3 \\\0 & 2 & 1 \\\4 & 1 & 1 \end{array}\right] \\\\\text { (c) }\left[\begin{array}{lll}1 & 2 & 1 \\\2 & 3 & 0 \\\0 & 1 & 2\end{array}\right]\end{array}$$

Find the determinant of the following matrices. (a) \(A=\left[\begin{array}{rr}1 & -34 \\ 0 & 2\end{array}\right]\) (b) \(A=\left[\begin{array}{rrr}4 & 3 & 14 \\ 0 & -2 & 0 \\ 0 & 0 & 5\end{array}\right]\) (c) \(A=\left[\begin{array}{rrrr}2 & 3 & 15 & 0 \\ 0 & 4 & 1 & 7 \\ 0 & 0 & -3 & 5 \\ 0 & 0 & 0 & 1\end{array}\right]\)

Suppose \(A, B\) are \(n \times n\) matrices and that \(A B=I .\) Show that then \(B A=I .\)

Show det \((a A)=a^{n} \operatorname{det}(A)\) for an \(n \times n\) matrix \(A\) and scalar a.

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?

An \(n \times n\) matrix is called nilpotent if for some positive integer, \(k\) it follows \(A^{k}=0 .\) If \(A\) is a nilpotent matrix and \(k\) is the smallest possible integer such that \(A^{k}=0,\) what are the possible values of \(\operatorname{det}(A) ?\)

Tell whether each statement is true or false. If true, provide a proof. If false, provide a counter example. (a) If \(A\) is a \(3 \times 3\) matrix with a zero determinant, then one column must be a multiple of some other column. (b) If any two columns of a square matrix are equal, then the determinant of the matrix equals zero. (c) For two \(n \times n\) matrices \(A\) and \(B, \operatorname{det}(A+B)=\operatorname{det}(A)+\operatorname{det}(B)\). (d) For an \(n \times n\) matrix \(A, \operatorname{det}(3 A)=3 \operatorname{det}(A)\) (e) If \(A^{-1}\) exists then \(\operatorname{det}\left(A^{-1}\right)=\operatorname{det}(A)^{-1}\). (f) If \(B\) is obtained by multiplying a single row of \(A\) by 4 then \(\operatorname{det}(B)=4 \operatorname{det}(A)\). (g) For A an \(n \times n\) matrix, \(\operatorname{det}(-A)=(-1)^{n} \operatorname{det}(A)\). (h) If \(A\) is a real \(n \times n\) matrix, then \(\operatorname{det}\left(A^{T} A\right) \geq 0\). (i) If \(A^{k}=0\) for some positive integer \(k,\) then \(\operatorname{det}(A)=0 .\) (j) If \(A X=0\) for some \(X \neq 0,\) then \(\operatorname{det}(A)=0 .\)