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Chapter 2: Problem 62

Let \(A\) be a nonsingular matrix. Prove that if \(B\) is row-equivalent to \(A,\) then \(B\) is also nonsingular.

Short Answer

Expert verified
If \(B\) is row-equivalent to a nonsingular matrix \(A\), \(B\) will also be nonsingular. None of the elementary row operations can make the determinant of a nonsingular matrix zero, thus \(B\) will also have a non-zero determinant.

Step by step solution

01

Understand Given

You are given two matrices \(A\) and \(B\). Matrix \(A\) is nonsingular which means it has a non-zero determinant. Matrix \(B\) is row-equivalent to \(A\) which means \(B\) has been obtained by a sequence of elementary row operations on \(A\).
02

Apply Elementary Row Operations

Elementary row operations include swapping two rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another row. None of these will result in a zero determinant if the original matrix had a non-zero determinant.
03

Prove B is Nonsingular

Since a sequence of elementary row operations on a nonsingular matrix will lead to another nonsingular matrix, we can say that \(B\), being row-equivalent to \(A\), is also nonsingular. Thus if \(B\) is row-equivalent to \(A\), then \(B\) is also nonsingular. The determinant of \(B\) will also be non-zero just like \(A\). Thus, truely \(B\) is nonsingular.

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