Lesson 3 · Examples and problems

Practice: Inverses, transposes and permutations

Three worked examples, then five problems. Check inverses by multiplying: A times your answer should give I.

Worked examples

Find the inverse of A = 4726.

det A = 4·6 − 7·2 = 10

A⁻¹ = 1/10 · 6−7−24

Check, top-left entry of AA⁻¹: (4·6 + 7·(−2)) / 10 = 1 ✓

Invert A = 1237 by elimination on [A | I].

[1 2 | 1 0; 3 7 | 0 1] → row 2 − 3·row 1 → [1 2 | 1 0; 0 1 | −3 1]

row 1 − 2·row 2 → [1 0 | 7 −2; 0 1 | −3 1]

A⁻¹ = 7−2−31

det A = 1, so the formula gives the same answer with no fractions.

Check (AB)ᵀ = BᵀAᵀ for A = 1201 and B = 1031.

AB = 7231, so (AB)ᵀ = 7321

BᵀAᵀ = 13011021 = 7321 ✓

AᵀBᵀ would be 1327: the wrong order gives a different matrix.

Find the inverse of 3152.

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det = 3·2 − 1·5 = 1, so A⁻¹ = 2−1−53.

Check: row 1 of A times column 1 of A⁻¹ = 3·2 + 1·(−5) = 1 ✓

For which k is 2k46 not invertible?

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det = 12 − 4k = 0 when k = 3. Then column 2 = (3, 6) = 1.5·column 1, and the plane is squashed onto a line.

A = 1101, B = 2003. Find (AB)⁻¹ as B⁻¹A⁻¹.

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B⁻¹ = ½00⅓ and A⁻¹ = 1−101.

B⁻¹A⁻¹ = ½−½0⅓. Check: AB = 2303 and its inverse from the formula, 1/6·3−302, is the same ✓

Compute AᵀA for A = 102103. Is it symmetric?

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AᵀA = 52210. Entry (1, 1) = 1 + 4 + 0, entry (1, 2) = 0 + 2 + 0, entry (2, 2) = 0 + 1 + 9.

Yes, it is symmetric, as AᵀA always is.

Find the permutation P that puts the rows of a 3 by 3 matrix in the order 2, 3, 1. What is P⁻¹?

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The rows of P are rows 2, 3, 1 of I: P = 010001100.

P⁻¹ = Pᵀ = 001100010, which puts the rows back.