Linear Algebra · Lesson 04

Chapter practice

Bring the chapter together: invert by formula, use the rules for products and transposes, read an inverse off the cofactors and solve with Cramer’s rule. No worked example sits above the answers.

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A A^{- 1} = I

(A B)^{- 1} = B^{- 1} A^{- 1}

det (A) \neq = 0

A^{- 1} = \frac{adj ( A )}{det ( A )}

A^{- 1} A = I

Chapter navigation

Intuition

Bring the chapter together: invert by formula, use the rules for products and transposes, read an inverse off the cofactors and solve with Cramer’s rule. No worked example sits above the answers.

Undoing is a skill with several tools, and the right one depends on the size of the job. These questions differ in which tool they need.

Before you practise

Decide first what is asked. A $2\times2$ inverse: the formula. A rule for a product, a power or a transpose: invert each factor and reverse the order. One entry of a larger inverse: a cofactor over the determinant. One unknown of a square system: Cramer's rule.

A^{-1}=\frac{1}{\det A}\operatorname{adj}A

Reminders

Invertible exactly when $\det A\neq0$ .
$(AB)^{-1}=B^{-1}A^{-1}$ and $(A^{T})^{-1}=(A^{-1})^{T}$ .
$(\operatorname{adj}A)_{ij}=C_{ji}$ : transpose the cofactors.
$x_i=\det A_i/\det A$ , with column $i$ replaced by .

Applications