A square matrix A of order n is invertible if there is a...

A square matrix $A$ of order $n$ is invertible if there is a matrix $B$ such that $A B=I_{n}=B A$ . Then $B$ is the inverse of $A,$ denoted by $A^{-1}$. Verify that $B=A^{-1}$ . Assume that $k=a d-b c \neq 0$ . $$A=\left[\begin{array}{rrr}{1} & {-2} & {0} \\ {3} & {1} & {-1} \\ {1} & {2} & {-3}\end{array}\right], B=\frac{1}{17}\left[\begin{array}{rrr}{1} & {6} & {-2} \\ {-8} & {3} & {-1} \\ {-5} & {4} & {-7}\end{array}\right]$$

A square matrix $A$ of order $n$ is invertible if there is a matrix $B$ such that $A B=I_{n}=B A$ . Then $B$ is the inverse of $A,$ denoted by $A^{-1}$. Verify that $B=A^{-1}$ . Assume that $k=a d-b c \neq 0$ .
$$A=\left[\begin{array}{rrr}{1} & {-2} & {0} \\ {3} & {1} & {-1} \\ {1} & {2} & {-3}\end{array}\right], B=\frac{1}{17}\left[\begin{array}{rrr}{1} & {6} & {-2} \\ {-8} & {3} & {-1} \\ {-5} & {4} & {-7}\end{array}\right]$$

Key Concepts

Identity Matrix

The identity matrix is a special square matrix with ones on the diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning that any matrix multiplied by the identity matrix (of suitable size) remains unchanged. This concept is essential in the definition of the inverse matrix.

Matrix Multiplication

Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. Understanding matrix multiplication is critical when verifying the inverse property since the multiplication of a matrix by its inverse must yield the identity matrix, demonstrating the fundamental relationships between the matrices.

Determinant

The determinant is a scalar value that can be computed from a square matrix and provides important information about the matrix, including whether it is invertible. A matrix is invertible if and only if its determinant is nonzero, making this concept central to the study of matrix invertibility.

Invertible Matrix

An invertible matrix, also known as a nonsingular matrix, is a square matrix that has an inverse. This means there exists another matrix such that when it is multiplied with the original matrix (from either side), the result is the identity matrix, which acts as the multiplicative identity in matrix algebra.

Matrix Inverse

The matrix inverse of an invertible matrix A is a matrix denoted by A?¹ such that A multiplied by A?¹ (in either order) gives the identity matrix. The computation and verification of an inverse are fundamental in solving systems of linear equations and in many areas of linear algebra.

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