If A and B are 3 ×3 matrices and |A| ≠0, then (a) |A B|=0 ⇒B |=0 (b) |A B| ≠0 ⇒B |≠0 (c) |A^-1|=

If A and B are 3 ×3 matrices and |A| ≠0, then (a) |A B|=0 ⇒B...

Key Concepts

Determinant of a Product

The determinant of a product of two matrices equals the product of their determinants. Mathematically, for any square matrices A and B, |AB| = |A||B|. This property is fundamental in analyzing how the invertibility or singularity of matrices behaves under multiplication.

Invertible Matrix

An invertible or non-singular matrix is one that has a nonzero determinant. This property implies that the matrix has an inverse. The condition |A| ? 0 guarantees that A is invertible, which in turn allows us to make further deductions about expressions involving A and its inverse.

Determinant of an Inverse Matrix

For any invertible matrix, the determinant of its inverse is the reciprocal of its determinant. In formal terms, if A is invertible, then |A^(-1)| = 1/|A|. This fact is derived from the multiplicative property of determinants and is crucial when working with matrix inverses.

Scalar Multiplication of Determinants

When a matrix is multiplied by a scalar, the determinant of the resulting matrix is the product of the scalar raised to the power of the matrix order (n) and the determinant of the original matrix. Specifically, for an n x n matrix A and scalar c, |cA| = c^n |A|. This concept explains how scaling a matrix affects its determinant and is particularly useful in problems where scalar multiples of matrices are considered.

Transcript

00:01 All right, dear learners, in this problem we are given two matrices, matrix a and matrix b.

00:11 Both are 3x3 matrixes.

00:14 And it is also given that the determinant of a is not equal to 0.

00:20 All right.

00:22 Now let's evaluate our options one by one.

00:25 Let's go for option number a.

00:27 It is given that the product of a and b whole determinant, is equal to 0 does that mean that determinant of b is equal to 0 alright let's see a b and determinant of a b times it can also be write that determinant of a times determinant of b we can write this as it we can write this thing like this as well is equal to 0 from here the product of two numbers is equal to 0 it means one of the numbers should be equal to to 0.

01:08 We know that a determinant is not equal to 0 because it is given in the question.

01:14 So therefore it means that b determinant should be equal to 0.

01:18 It means this option is correct.

01:23 All right.

01:24 Now let's move to option number b.

01:28 In option b it is given that the determinant of a times b is not equal to 0.

01:40 Thus this means date b determinant is not equal to 0.

01:47 All right.

01:49 Let's evaluate this.

01:51 This can also be written like this.

01:56 Determinant of a times determinant of b is not equal to 0...