Prove that if A is an invertible matrix, then A A^T and A^T A are also invertible.

Name: Problem 9, Chapter 1: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-06-18T06:29:39-08:00
Duration: 4 min 9 s
Channel: Anthony Ramos
Description: Prove that if A is an invertible matrix, then A A^T and A^T A are also invertible.

step 1 Hi, so in this occasion we need to prove this statement here. So let's suppose that a is a squar

Prove that if A is an invertible matrix, then A A^T and A^T...

Key Concepts

Product of Invertible Matrices

The product of two invertible matrices is itself invertible, and the inverse of the product is given by the product of the inverses in the reverse order. This concept is critical when analyzing products like A A?, as it allows one to directly construct the inverse of the product using the inverses of the original matrices, provided they exist.

Determinant and Invertibility

The determinant of a matrix is a scalar value that provides important information about the matrix, including whether it is invertible. A key property is that the determinant of a product of matrices equals the product of their determinants. For an invertible matrix A, the determinant is non-zero, and hence the determinants of A A? or A? A are also non-zero (in fact, they equal the square of the determinant of A), confirming the invertibility of these products.

Invertible Matrix

An invertible matrix is a square matrix that has a multiplicative inverse, meaning there exists another matrix which, when multiplied with the original, yields the identity matrix. This property is essential because it guarantees that the matrix has full rank and a non-zero determinant, which are critical in solving systems of linear equations and understanding linear transformations.

Matrix Transpose

The transpose of a matrix is obtained by interchanging its rows and columns. In the context of invertible matrices, an important property is that the transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. This property is often used to simplify proofs and computations involving matrix products that include transposed matrices.

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