91Ó°ÊÓ

We are given five statements and we need to determine whether each one is true or false. For each true statement, we need to provide a reason, and for each false statement, we need to provide a counterexample.

The statement is that if matrices \(A\) and \(B\) are identical except that \(b_{11}=2a_{11}\), then \(\operatorname{det} B=2 \operatorname{det} A\). This is false. A counterexample can be two 2x2 matrices: \(A=\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\) and \(B=\begin{pmatrix} 2 & 0 \ 0 & 1 \end{pmatrix}\). Their determinants are \(\operatorname{det} A = 1\) and \(\operatorname{det} B = 2\), which satisfy \(\operatorname{det} B = 2 \operatorname{det} A\); however, try \(A=\begin{pmatrix} 1 & 3 \ 1 & 1 \end{pmatrix}\) and \(B=\begin{pmatrix} 2 & 3 \ 1 & 1 \end{pmatrix}\). With these choice, \(\operatorname{det} A eq 1\) and is not half the determinant of \(B\). Hence, the pattern when scaling just one element doesn't hold universally.

The determinant is indeed the product of the pivots after the matrix is transformed into upper triangular form through row operations. This is true. In the process of row reduction (specifically LU decomposition), the pivots (elements on the diagonal of an upper triangular matrix \(U\)) multiply to give the determinant.

For \(A\) invertible and \(B\) singular, \(A+B\) could be invertible or not, depending on the matrices. For \(A=\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\) and \(B=\begin{pmatrix} -1 & 0 \ 0 & 0 \end{pmatrix}\), \(A + B = \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}\) is singular. Therefore, this statement is false.

If \(A\) is invertible and \(B\) is singular, \(AB\) must be singular. From matrix properties, if \(B\) is singular, its determinant is zero. The determinant of a product is the product of determinants, so \(\det(AB) = \det(A)\det(B) = 0\). This makes \(AB\) singular. Therefore, this statement is true.

The determinant of \(AB - BA\) being zero is true. If \(A\) and \(B\) are compatible square matrices, then \(AB - BA\) is known as a commutator which always has a trace of zero. In particular, for any matrices \(A\) and \(B\), the determinant of \(AB - BA\) equates to zero due to its skew-symmetrical properties in terms of invariant operations.