Matrix Algebra Lab: Determinants and Inverses

MTH-168 · Semester II · Mathematics II

intermediate25 minAvailable

Limits of quotients

Guided practice — solve each problem, check, then generate a fresh one.

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Formula to apply

lim_{x→a} (x² − a²)/(x − a) = 2a

Problem

Evaluate lim_{x→5} (x² − 25) / (x − 5).

limit

Working

Lab brief

Compute determinants and inverse matrices for 2×2 matrices, using the adjugate formula A⁻¹ = (1/det)[[d, −b], [−c, a]].

Theory & concepts

A square matrix is invertible exactly when its determinant is non-zero. For a 2×2 matrix the inverse has a clean closed form.

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Basic 2x2 matrix operations

Adjugate formula

If A = [[a, b], [c, d]] then A⁻¹ = (1/(ad − bc))·[[d, −b], [−c, a]]. Swap the diagonal, negate the off-diagonal, divide by the determinant.

When det = ±1 the inverse has integer entries — a favourite of exam setters and a quick self-check: multiplying A·A⁻¹ must give the identity matrix.

Check with the identity

A·A⁻¹ = I. If your product is not the identity, at least one of the entries is wrong.

Invert A = [[2, 3], [1, 2]]
det = 1, so A⁻¹ = [[2, −3], [−1, 2]].
Verify with the identity
Multiply your inverse by A and confirm the product is the 2×2 identity.
Non-invertible matrix
Construct a 2×2 matrix with determinant zero and explain why it has no inverse.

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