Determinant and inverse of a matrix

Computing a determinant (2 × 2, Sarrus, expansion along a row, triangular), deciding whether a matrix is invertible, computing its inverse and solving AX = B with it.

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Determinant: what is it?
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A number attached to a square matrix, equal to ad − bc for the matrix (a b; c d), and zero exactly when the matrix is not invertible.
For (2 1; 5 3), 2×3 − 1×5 = 1: nonzero, so the matrix is invertible. Up to sign, the determinant is the factor by which the associated transformation of the plane scales areas.
What is the determinant of the matrix (a b; c d)?
- ad − bc
- ad + bc
- ac − bd
- ab − cd
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ad − bc
Multiply along the diagonals: a×d for the main diagonal, b×c for the other, then subtract. The decoys mix up positions or change the sign.
A square matrix A is invertible if and only if its determinant is ….
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A square matrix A is invertible if and only if its determinant is nonzero.
It is the fastest test for the invertibility of a small matrix. A determinant of 1 or −1 is nonzero like any other.
Invertible matrix: what is it?
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A square matrix A for which there is a matrix B of the same size such that AB = BA = I.
It corresponds to a system AX = B that always has exactly one solution. Only square matrices can be, but not every square matrix is.

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