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Inverse matrices · Linear Algebra | Math Infinitum

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Linear Algebra
Inverse matrices

Linear Algebra · Chapter 05

Inverse matrices

When a square matrix can be undone, and how to write its inverse down: the 2 by 2 formula, the rules inverses obey, the adjugate and Cramer’s rule.

When a square matrix can be undone, and how to write its inverse down: the 2 by 2 formula, the rules inverses obey, the adjugate and Cramer’s rule.

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

x = A^{- 1} b

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

x_{i} = \frac{det ( A _{i} )}{det ( A )}

Lessons

Build the foundation

01Inverting a 2×2 matrixA 2 by 2 matrix can be inverted by a formula: swap the two diagonal entries, change the signs of the other two, and divide by the determinant. The division is exactly where things break when the determinant is zero.Read lesson →
02Rules for inversesInverses follow a few rules that come straight from undoing things in order. Undoing two steps means undoing the last one first, so the inverse of a product reverses the order of its factors.Read lesson →
03The adjugate and Cramer’s ruleThe cofactors of a matrix, arranged the right way, almost give its inverse. Put them in a matrix, transpose it, and divide by the determinant. The same idea solves a square system one unknown at a time, each unknown a ratio of two determinants.Read lesson →
04Chapter practiceBring 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.Read lesson →