91Ó°ÊÓ

Prove the associative property of matrix addition: \(A+(B+C)=(A+B)+C\). Getting Started: To prove that \(A+(B+C)\) and \((A+B)+C\) are equal, show that their corresponding entries are equal. (i) Begin your proof by letting \(A, B,\) and \(C\) be \(m \times n\) matrices. (ii) Observe that the \(i j\) th entry of \(B+C\) is \(b_{i j}+c_{i j^{*}}\) (iii) Furthermore, the \(i j\) th entry of \(A+(B+C)\) is \(a_{i j}+\left(b_{i j}+c_{i j}\right)\) (iv) Determine the ijth entry of \((A+B)+C\)

Show that the matrix below does not have an \(L U\) -factorization. \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\)

Prove Property 4 of Theorem 2.8: If \(A\) is an invertible matrix, then \(\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}\)

Proof Prove that each statement is true when \(A\) and \(B\) are square matrices of order \(n\) and \(c\) is a scalar. (a) \(\operatorname{Tr}(A+B)=\operatorname{Tr}(A)+\operatorname{Tr}(B)\) (b) \(\operatorname{Tr}(c A)=c \operatorname{Tr}(A)\)

Prove that if \(A^{2}=A,\) then either \(A\) is singular or \(A=I\) Getting Started: You must show that either \(A\) is singular or \(A\) equals the identity matrix. (i) Begin your proof by observing that \(A\) is either singular or nonsingular. (ii) If \(A\) is singular, then you are done. (iii) If \(A\) is nonsingular, then use the inverse matrix \(A^{-1}\) and the hypothesis \(A^{2}=A\) to show that \(A=I\)

Is the sum of two invertible matrices invertible? Explain why or why not. Illustrate your conclusion with appropriate examples.

Writing Under what conditions will the diagonal matrix $$A=\left[\begin{array}{ccccc}a_{11} & 0 & 0 & \ldots & 0 \\ 0 & a_{22} & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \ldots & a_{n n}\end{array}\right]$$ be invertible? Assume that \(A\) is invertible and find its inverse.

Determine whether the matrix is symmetric, skew-symmetric, or neither. A square matrix is skew-symmetric when \(A^{T}=-A\). $$ A=\left[\begin{array}{rrr}0 & 2 & -1 \\\\-2 & 0 & -3 \\\1 & 3 & 0\end{array}\right] $$

Proof Prove that if both products \(A B\) and \(B A\) are defined, then \(A B\) and \(B A\) are square matrices.

Prove that the main diagonal of a skew-symmetric matrix consists entirely of zeros.