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Chapter 3: Problem 49

Let \(A\) and \(B\) be \(n \times n\) matrices such that \(A B=I .\) Prove that \(|A| \neq 0\) and \(|B| \neq 0\)

Short Answer

Expert verified
The determinants of matrices A and B are not equal to zero, i.e., \(|A| \neq 0\) and \(|B| \neq 0\). This conclusion is based on the property that the determinant of the product of matrices equals the product of their determinants and the fact that multiplication of A and B results in the identity matrix.

Step by step solution

01

Understanding the problem

As per the problem, the matrix multiplication AB is equal to the identity matrix I i.e., \(AB = I\). This implies that B is the inverse of matrix A, i.e., \(B = A^{-1}\). Similarly, A is the inverse of B. Now the task here is to prove that the determinants of matrices A and B are not equal to zero.
02

Apply properties of determinant

The task can be accomplished by using the properties of determinants. One suitable property here would be: 'The determinant of a product of matrices is the product of their determinants'. So, if \(AB = I\), then \(|AB| = |I|\). This means that \(|A|\cdot |B| = 1\) (because the determinant of the identity matrix is 1).
03

Final Analysis - Determinant of A and B

From Step 2, it's clear that the product \(|A|\cdot |B| = 1\). From this it can be inferred that neither \(|A|\) nor \(|B|\) can be zero, because if either of them is zero the equation wouldn't hold true as the product would be zero. Hence, this proves that for matrices A and B, their determinant can't be zero i.e., \(|A| \neq 0\) and \(|B| \neq 0\).

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