Inverses and matrix problems

For numbers, the inverse of 5 is 1/5, because 5 × 1/5 = 1. For a square matrix A, an inverse is a matrix B with AB = BA = I. We write it A⁻¹.

• Only square matrices can have inverses.
• Not every square matrix has one (just as 0 has no inverse).
• If an inverse exists, it is unique: there is only one.
• (AB)⁻¹ = B⁻¹A⁻¹: the order reverses, as with the transpose.

Checking a claimed inverse is easy: multiply. If AB = I and BA = I, B is the inverse. The next chapter shows how to find inverses.

Finding an unknown matrix. First decide its order from the multiplication rule, then give its entries letters and compare entries. Example: if X[1, 2] = [3, 6] and X is 1 × 1, then X = [3].

Simplifying with powers. Treat the identity I like the number 1: I² = I, I³ = I, and IA = AI = A, so I commutes with everything. Then the usual algebra works.

Example: if A² = A, then (I + A)² = I² + 2IA + A² = I + 2A + A = I + 3A. So (I + A)² – 3A = I.

Be careful: (A + B)² = A² + AB + BA + B², which equals A² + 2AB + B² only when AB = BA.

Word problems. Put the data in matrices so that the rows of the first match the columns of the second. If a shop sells (10, 6, 4) items of three kinds in a day and they cost ₹20, ₹50 and ₹30, the takings are the product [10, 6, 4][20; 50; 30] = [200 + 300 + 120] = [620], so ₹620.