Use part (c) of the Invertible Matrix Theorem to prove that...

Use part (c) of the Invertible Matrix Theorem to prove that if $A$ is an invertible matrix and $B$ and $C$ are matrices of the same size as $A$ such that $A B=A C$, then $B=C .$ IHint: Consider $A B-A C=0.1.$

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Key Concepts

Invertible Matrix

An invertible matrix is a square matrix that has a multiplicative inverse. This means there exists another matrix such that when the two are multiplied together, the result is the identity matrix. The existence of an inverse is the key property that allows for solving matrix equations and performing algebraic manipulations such as cancellation.

Matrix Inverse

The inverse of an invertible matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is used to 'reverse' the operation of a matrix, enabling one to isolate and solve for unknown matrices in an equation.

Identity Matrix

The identity matrix is a special square matrix with ones on the diagonal and zeros elsewhere. It serves as the neutral element in matrix multiplication, meaning that any matrix multiplied by the identity matrix remains unchanged. This property is fundamental when using inverses to simplify matrix equations.

Cancellation Law in Matrix Multiplication

This law states that if an invertible matrix multiplies two matrices to yield the same result, then the two matrices must be equal. Because an invertible matrix can be 'canceled' from both sides of the equation using its inverse, it becomes a powerful tool in establishing the equality of matrices in matrix equations.

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