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Proof: 2x2 matrix invertible iff det not equal to 0

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Please prove the following statement:

The matrix: a b
c d
is invertible if and only if ad-bc does not equal zero

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Solution Summary

This solution provides proof that invertible implies that the determinant is nonzero. Additionally, it provides proof that nonzero determinant implies invertibility. The solution provides two examples of these proofs.

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Proof happens in two steps.
1. Assume that the matrix is invertible and show that ad-bc is not zero.
2. Assume that ad-bc is not zero then show that the matrix is invertible.

Note:
a. To be invertible means that the inverse of the matrix exists i.e. A *A^-1 = I
b. ad-bc is the determinant of the matrix.

So for step 1. Assume A*A^-1 = I. Then use two properties of the determinant i. det(A*B) = det(A) *det (B) and ii. det (I) = 1. This way the det(A) could not be zero.

For step 2. Assume that det(A) is nonzero. There is a quick way to find the inverse of 2x2 matrix A.
A^-1 = [1/det(A)]*[d -b
-c a] That is switch ...

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