91Ó°ÊÓ

Let \(A\) be a nonsingular \(n \times n\) matrix with a nonzero cofactor \(A_{m n},\) and set \\[ c=\frac{\operatorname{det}(A)}{A_{n n}} \\] Show that if we subtract \(c\) from \(a_{n n},\) 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 \\[ \begin{array}{c} E=\left(\begin{array}{cc} I_{k} & O \\ O & B \end{array}\right), \quad F=\left(\begin{array}{cc} A & O \\ O & I_{n-k} \end{array}\right) \\ C=\left(\begin{array}{cc} A & O \\ O & B \end{array}\right) \end{array} \\] where \(I_{k}\) and \(I_{n-k}\) are the \(k \times k\) and \((n-k) \times(n-k)\) identity matrices. (a) Show that det( \(E)=\operatorname{det}(B)\) (b) Show that \(\operatorname{det}(F)=\operatorname{det}(A)\) (c) Show that \(\operatorname{det}(C)=\operatorname{det}(A) \operatorname{det}(B)\).

Let \(A\) and \(B\) be \(k \times k\) matrices and let \\[ M=\left(\begin{array}{ll} O & B \\ A & O \end{array}\right) \\] Show that \(\operatorname{det}(M)=(-1)^{k} \operatorname{det}(A) \operatorname{det}(B)\).

Show that evaluating the determinant of an \(n \times n\) matrix by cofactors involves \((n !-1)\) 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 \([n(n-1)(2 n-1)] / 6\) additions and \(\left[(n-1)\left(n^{2}+n+3\right)\right] / 3\) multiplications and divisions. Hint: At the 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 \(n\).