HW 5: Let A be an invertible square matrix. If AB = AC, then B = C.
TRUE. AB = AC–> A^−1 (AB) = A^−1 (AC) –> InB = InC –> B = C
Let us begin the proof by assuming that A is an invertible square matrix, and AB = AC. We want to prove that B = C.
We can start by multiplying both sides of the equation AB = AC by A inverse on the left. This gives:
A inverse AB = A inverse AC
Now, since A is invertible, we can cancel out A inverse on both sides of the equation to get:
B = C
Therefore, we have proved that if AB = AC and A is an invertible square matrix, then B = C. This result is important because it tells us that if we want to solve an equation of the form AB = AC, where A is an invertible matrix, we can simply multiply both sides by A inverse and cancel out A inverse to obtain the solution.
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