True or false: If true, explain why. If false, give an explicit counterexample. (a) If det A ≠0

True or false: If true, explain why. If false, give an...

Key Concepts

Determinant and Invertibility

A square matrix is invertible if and only if its determinant is nonzero. This is a fundamental property which connects the determinant of a matrix with the existence of an inverse, implying that a nonzero determinant ensures the linear independence of the columns (or rows) and full rank of the matrix.

Scalar Multiplication Properties of Determinants

When a matrix is multiplied by a scalar, the determinant of the resulting matrix is the scalar raised to the power of the matrix order times the determinant of the original matrix. This scaling property is crucial in understanding how uniform scaling of all elements of a matrix affects its volume-scaling factor.

Non-Additivity of the Determinant

Unlike many linear operations, the determinant is not an additive function. That is, the determinant of a sum of matrices is not generally equal to the sum of their determinants. This non-additivity underlines the non-linear nature of the determinant function and emphasizes caution when dealing with operations that involve matrix summations.

Determinant of Transpose and Inverse Matrices

The determinants of a matrix and its transpose are equal, and when a matrix is invertible, the determinant of its inverse is the reciprocal of the determinant of the original matrix. These properties are instrumental in various proofs and applications where transposition and inversion interact.

Multiplicative Property of Determinants

One of the key properties of determinants is that the determinant of a product of matrices is equal to the product of their individual determinants. This multiplicative property allows for simplification of expressions involving the determinants of complex products and is widely used in proofs and computations.

Matrix Non-Commutativity and Factorization

Matrix multiplication is generally non-commutative, which affects how algebraic identities, such as factorization formulas, translate when dealing with matrices. Recognizing that factors may not commute is essential when attempting to extend scalar identities, like the difference of squares, to matrix equations.

Rank, Singularity, and Determinant

The determinant of a matrix provides insight into its rank; a zero determinant signifies that the matrix is singular, meaning it does not have full rank. This directly relates to the linear dependence of rows or columns and the inability of the matrix to span its entire vector space.

Invertibility and Zero Product Property

If a matrix with a nonzero determinant (hence invertible) multiplies another matrix to give the zero matrix, then the other matrix must also be the zero matrix. This property stems from the fact that invertible matrices do not have nontrivial kernels, which is a key concept in understanding solutions to matrix equations.

Transcript

00:02 In problem 14, we have to check whether the given statements are true or false.

00:11 So first consider the statement in part a of this problem.

00:18 Now, to check the truth of the statement, let us consider two matrices satisfying the given condition.

00:29 That is matrix a is equal to, 2 to 2 and matrix b is equal to 4 .2.

00:53 Here we see that the entry, here is the entry a11 and entry a1 of the first matrix and entry b11 of the s3b1 of the 4 is the entry b11 of the second matrix and 2 is the entry a11 of the first matrix.

01:24 Clearly, a11 is equal to 2 times a11.

01:33 Now we are going to find determinants of the matrix a and b.

01:40 Clearly, we see that determinant of a is equal to 2 times 2 is 4 minus 2 times 2 is 4 is equal to 0, and determinant of b is equal to 4 times 2 is 8 minus 2 times 2 is 4.

02:02 That is determinant of a is equal to 0, and determinant of the matrix b, is equal to four.

02:14 That is determinant of the matrix b is not equal to two times determinant of matrix a.

02:35 Thus, the given statement is false.

02:44 Now we are going to check the statement b.

02:48 For this, the determinant of the permutation matrix that is determinant of a is equal to determinant of 0 -0 -0 -1 -1 -0 -0 -0 -0 -1 -0 -1 -0 -1 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0.

03:24 Now, we are going to evaluate this determinant using row operations.

03:37 So interchanging row 1 and row 4, we get this determinant is equal to 1 ,000, 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -1.

03:59 And 1 and 0 1 0 by interchanging row 1 and row 4.

04:19 From the properties of determinant, we know that if we interchange any two rows, then the sign of determinant will be negative.

04:33 So this is equal to minus 1 power 2, that is we have to multiply it with by another negative sign.

04:50 When we interchange row 2 and row 4 of this determinant above determinant that is 1 000 0100 0100 0100 0 0 0 0 0 1 1 1 and 0 0 0 0 0 0 0 0 0 0 0.

05:14 Here we interchange row 2 and row 4 4.

05:21 Now, if we interchange row 2 and row 3, we have to multiply with another minus 1, so we have minus 1 cube, and the determinant will be 1 -0 -0 -1 -0 -0 -0 -0 -0 -0 -0 -0 -1 -0 -0 -0 -1 -0 and 0 -0 -0 -0 -0 -0 -1, and 0 -0 -0 -0 -0 -0 -1.

05:52 Row 1 by interchanging row 2 and row 3.

06:02 So as we know that determinant of identity matrix is equal to 1 because the product of the entries of pivots is equal to 1.

06:15 So the determinant will be minus 1 cube times 1 is equal to minus 1.

06:25 One cube is minus 1.

06:27 So minus 1 times 1 is equal to minus 1...

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