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Chapter 1: Problem 26

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

Short Answer

Expert verified
In summary, we have proved that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B = MA. This was shown by proving (1) if B is row equivalent to A, then there exists a nonsingular matrix M, and (2) if there exists a nonsingular matrix M such that B = MA, then B is row equivalent to A. Thus, our claim is established.

Step by step solution

01

1. Proving B is row equivalent to A implies a nonsingular matrix M exists'

If B is row equivalent to A, then B can be obtained by performing a series of elementary row operations on A. Since elementary row operations can be represented by multiplication with elementary matrices, this means there exists a matrix M that can be obtained by multiplying a series of elementary matrices, such that B = MA. Elementary matrices are nonsingular because their determinants are nonzero (they are only obtained by performing elementary row operations on the identity matrix). Therefore, the product of elementary matrices is also nonsingular. Thus, there exists a nonsingular matrix M such that B = MA.
02

2. Proving the existence of nonsingular matrix M implies B is row equivalent to A'

If there exists a nonsingular matrix M such that B = MA, this implies that B can be obtained by multiplying A with a series of elementary matrices (since every nonsingular matrix can be factored into a product of elementary matrices). Multiplying A with elementary matrices corresponds to performing a series of elementary row operations on A. Therefore, B can be obtained by performing a series of elementary row operations on A, which implies that B is row equivalent to A. Now we have proved both (1) and (2), so we have shown that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B = MA.

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Most popular questions from this chapter

For each of the following pairs of matrices, find an elementary matrix \(E\) such that \(A E=B:\) (a) \(A=\left(\begin{array}{lll}4 & 1 & 3 \\ 2 & 1 & 4 \\ 1 & 3 & 2\end{array}\right), B=\left(\begin{array}{lll}3 & 1 & 4 \\ 4 & 1 & 2 \\ 2 & 3 & 1\end{array}\right)\) (b) \(A=\left(\begin{array}{ll}2 & 4 \\ 1 & 6\end{array}\right), B=\left(\begin{array}{rr}2 & -2 \\ 1 & 3\end{array}\right)\) (c) \(A=\left(\begin{array}{rrr}4 & -2 & 3 \\ -2 & 4 & 2 \\ 6 & 1 & -2\end{array}\right)\) \(B=\left(\begin{array}{rrr}2 & -2 & 3 \\ -1 & 4 & 2 \\ 3 & 1 & -2\end{array}\right)\)

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 \\[ A=\left(\begin{array}{cc} A_{11} & A_{12} \\ O & A_{22} \end{array}\right) \\] where all four blocks are \(n \times n\) matrices. (a) If \(A_{11}\) and \(A_{22}\) are nonsingular, show that \(A\) must also be nonsingular and that \(A^{-1}\) must be of the form \\[ \left(\begin{array}{c|c} A_{11}^{-1} & C \\ \hline O & A_{22}^{-1} \end{array}\right) \\] (b) Determine \(C\)

Given \\[ X=\left(\begin{array}{lll} 2 & 1 & 5 \\ 4 & 2 & 3 \end{array}\right) \quad Y=\left[\begin{array}{lll} 1 & 2 & 4 \\ 2 & 3 & 1 \end{array}\right] \\] (a) Compute the outer product expansion of \(X Y^{T}\) (b) Compute the outer product expansion of \(Y X^{T}\) How is the outer product expansion of \(Y X^{T}\) related to the outer product expansion of \(X Y^{T} ?\)

Let \\[ A=\left(\begin{array}{ll} O & I \\ B & O \end{array}\right) \\] where all four submatrices are \(k \times k\). Determine \(A^{2}\) and \(A^{4}\)
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