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Chapter 8: Problem 27

Suppose \(B\) is row equivalent to a square matrix \(A\). Prove that \(|B|=0\) if and only if \(|A|=0\).

Short Answer

Expert verified
We are given that matrix $B$ is row equivalent to matrix $A$. Two matrices are row equivalent if one can be obtained from the other by performing a finite number of elementary row operations. We have shown that for each type of elementary row operation (swapping rows, row multiplication by a non-zero scalar, and adding/subtracting a multiple of a row to another row), the if and only-if relation holds between the determinants of the matrices. Hence, if $B$ is row equivalent to a square matrix $A$, then \(|B|=0\) if and only if \(|A|=0\).

Step by step solution

01

Defining Row Equivalence and Elementary Row Operations

Two matrices A and B are said to be row equivalent if one can be obtained from the other by performing a finite number of elementary row operations. There are three types of elementary row operations: 1. Swapping two rows. 2. Multiplication of a row by a non-zero scalar. 3. Adding or subtracting a multiple of one row to another row.
02

Determinant Property for Swapping Rows

When we swap two rows of a square matrix, the determinant of the resulting matrix is the negative of the original determinant. If A and B are row equivalent, and B can be obtained from A by swapping rows, then \(|B| = -|A|\).
03

Determinant Property for Row Multiplication

When we multiply a row of a square matrix by a non-zero scalar, the determinant of the resulting matrix is also multiplied by that scalar. If A and B are row equivalent, and B can be obtained from A by multiplying a row by a scalar k, then \(|B| = k|A|\).
04

Determinant Property for Adding/Subtracting Rows

When we add or subtract a multiple of one row to another row in a square matrix, the determinant of the resulting matrix remains unchanged. If A and B are row equivalent, and B can be obtained from A by adding/subtracting a multiple of a row to another row, then \(|B| = |A|\).
05

Proving the Relation Between Determinants of Row Equivalent Matrices

Let's assume B is row equivalent to A and B can be obtained from A by performing some elementary row operations. Case 1: If the operation is swapping rows, then \(|B| = -|A|\). In this case, if \(|A| = 0\), then \(|B| = -0 = 0\). And if \(|B| = 0\), then \(|A| = -0 = 0\). Case 2: If the operation is row multiplication by a non-zero scalar k, then \(|B| = k|A|\). In this case, if \(|A| = 0\), then \(|B| = k(0) = 0\). And if \(|B| = 0\), then \(0 = k|A|\), which implies \(|A| = 0 \) as k is not zero. Case 3: If the operation is adding/subtracting a multiple of a row to another row, then \(|B| = |A|\). In this case, if \(|A| = 0\), then \(|B| = 0\). And if \(|B| = 0\), then \(|A| = 0\). Since B can be obtained from A through a finite sequence of elementary row operations, the if and only if relation holds for each operation and thus holds for the entire sequence of operations. Hence, we have shown that if B is row equivalent to a square matrix A, then \(|B| = 0\) if and only if \(|A| = 0\).

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