Q:

Construct a 3×3 nonsingular matrix and call it A

Nesma
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2022-05-28
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Construct a 3×3 nonsingular matrix and call it A.   Then, for each entry of the matrix, compute the corresponding cofactor, and create a new 3 ×3 matrix full of these cofactors by placing the cofactor of an entry in the same location as the entry it was based on. Once complete, call this matrix C. Compute ACt. Any observations? Repeat with a new matrix, or perhaps with a 4 × 4 matrix.

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Nesma
2022-05-28
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The result of these computations should be a matrix with the value of det (A) in the diagonal entries and zeros elsewhere. The suggestion of using a nonsingular matrix was partially so that it was obvious that the value of the determinant appears on the diagonal.

This result (which is true in general) provides a method for computing the inverse of a nonsingular ma- trix. Since ACt = det (A) In, we can multiply by the reciprocal of the determinant (which is nonzero!) and the inverse of A (it exists!) to arrive at an expression for the matrix inverse:

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