91Ó°ÊÓ

Let \(A\) be a \(2 \times 2\) matrix with \(a_{11} \neq 0\) and let \(\alpha=a_{21} / a_{11} .\) Show that \(A\) can be factored into a product of the form $$\left(\begin{array}{ll} 1 & 0 \\ \alpha & 1 \end{array}\right)\left(\begin{array}{cc} a_{11} & a_{12} \\ 0 & b \end{array}\right)$$ What is the value of \(b ?\)

Let \(U\) be an \(n \times n\) upper triangular matrix with nonzero diagonal entries. (a) Explain why \(U\) must be nonsingular (b) Explain why \(U^{-1}\) must be upper triangular.

Let \(A\) be an \(n \times n\) matrix. Show that if \(A^{k+1}=O\) then \(I-A\) is nonsingular and $$(I-A)^{-1}=I+A+A^{2}+\cdots+A^{k}$$

Let \(A\) be a nonsingular \(n \times n\) matrix and let \(B\) be an \(n \times r\) matrix. Show that the reduced row echelon form of \((A | B)\) is \((I | C),\) where \(C=A^{-1} B\).

Prove that if \(A\) is row equivalent to \(B\) then \(B\) is row equivalent to \(A\)

(a) Prove that if \(A\) is row equivalent to \(B\) and \(B\) is row equivalent to \(C,\) then \(A\) is row equivalent to \(C\) (b) Prove that any two nonsingular \(n \times n\) matrices are row equivalent

Let \(A\) be an idempotent matrix. (a) Show that \(I-A\) is also idempotent. (b) Show that \(I+A\) is nonsingular and \((I+A)^{-1}=I-\frac{1}{2} A\)

Prove that \(B\) is row equivalent to \(A\) if and only if there exists a nonsingular matrix \(M\) such that \(B=M A\)

Is it possible for a singular matrix \(B\) to be row equivalent to a nonsingular matrix \(A\) ? Explain.

Let \(A\) be an \(m \times n\) matrix. Show that \(A^{T} A\) and \(A A^{T}\) are both symmetric.