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 the same. (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 \(i j\) th entry of \((A+B)+C\)

(a) Show that the matrix \\[ A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] \\] does not have an \(L U\) -factorization. (b) Find the \(L U\) -factorization of the matrix \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) that has 1 's along the main diagonal of \(L\). Are there any restrictions on the matrix \(A\) ?

Prove the associative property of scalar multiplication: \((c d) A=c(d A)\)

Let \(A\) and \(B\) be \(3 \times 3\) matrices, where \(A\) is diagonal. (a) Describe the product \(A B\). Illustrate your answer with examples. (b) Describe the product \(B A\). Illustrate your answer with examples. (c) How do the results in parts (a) and (b) change if the diagonal entries of \(A\) are all equal?

Prove that the inverse of a symmetric nonsingular matrix is symmetric. Getting Started: To prove that the inverse of \(A\) is symmetric, you need to show that \(\left(A^{-1}\right)^{T}=A^{-1}\). (i) Let \(A\) be a symmetric, nonsingular matrix. (ii) This means that \(A^{T}=A\) and \(A^{-1}\) exists. (iii) Use the properties of the transpose to show that \(\left(A^{-1}\right)^{T}\) is equal to \(A^{-1}\)

Prove that the scalar 1 is the identity for scalar multiplication: \(1 A=A\)

Prove that if \(A^{2}=A\), then either \(A=I\) or \(A\) is singular 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 \(A=I\)

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

Prove that \(A\) is idempotent if and only if \(A^{T}\) is idempotent. Getting Started: The phrase "if and only if" means that you have to prove two statements: 1\. If \(A\) is idempotent, then \(A^{T}\) is idempotent. 2\. If \(A^{T}\) is idempotent, then \(A\) is idempotent. (i) Begin your proof of the first statement by assuming that \(A\) is idempotent. (ii) This means that \(A^{2}=A\) (iii) Use the properties of the transpose to show that \(A^{T}\) is idempotent. (iv) Begin your proof of the second statement by assuming that \(A^{T}\) is idempotent.

Let \(A\) and \(B\) be two matrices such that the product \(A B\) is defined. Show that if \(A\) has two identical rows, then the corresponding two rows of \(A B\) are also identical.