4.9 The Inverse and Determinant of a Matrix

Determinant of a 2×2 Matrix

For matrix A = [[a,b],[c,d]], the determinant is det(A) = ad − bc. The determinant tells whether a matrix has an inverse: if det(A) = 0, the matrix is singular (no inverse exists). Geometrically, the determinant relates to the scaling factor of the area transformation applied by the matrix.

Finding the Inverse

The inverse of A = [[a,b],[c,d]] is A⁻¹ = (1/det(A)) × [[d,−b],[−c,a]], provided det(A) ≠ 0. Multiplying a matrix by its inverse gives the identity matrix I = [[1,0],[0,1]]: A·A⁻¹ = A⁻¹·A = I.

Using Inverses to Solve Systems

A system of linear equations can be written as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constants matrix. If A⁻¹ exists, solve with X = A⁻¹B. This method only works when det(A) ≠ 0, meaning the system has a unique solution.

Key terms

Determinant of [[a,b],[c,d]]

det(A) = ad - bc

Inverse of a 2x2 matrix

A⁻¹ = (1/det(A)) × [[d,-b],[-c,a]], only if det(A) ≠ 0

Singular matrix

A matrix with determinant equal to 0; it has no inverse.

Identity matrix

The matrix I such that A·I = I·A = A; for 2x2, I = [[1,0],[0,1]].

Solving AX = B using inverses

If A⁻¹ exists, then X = A⁻¹B gives the unique solution to the system.

Try an AP-style question

det [[3, 1], [2, 4]] = ?

1. 12
2. 10
3. 2
4. 14