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When you multiply two matrices in Class 12 and get a different answer depending on the order, you are meeting an idea that was new in 1858. That year Arthur Cayley, then a London lawyer, published a paper that set out matrix algebra as a subject of its own. This post covers his life, what the paper did, and three worked examples that show his ideas at Class 12 level.
Who was Arthur Cayley?
Cayley was born in Richmond, Surrey, on 16 August 1821 and died in Cambridge on 26 January 1895, according to his MacTutor biography. He entered Trinity College, Cambridge in 1838 and graduated as Senior Wrangler, the top mathematics student, in 1842.
His Cambridge fellowship could not last, so he trained in law and was admitted to the bar on 3 May 1849. He spent 14 years as a lawyer and published about 250 mathematical papers in that time. MacTutor records his remark that the object of law was to say a thing in the greatest number of words, and of mathematics to say it in the fewest. In 1863 he became Sadleirian Professor of Pure Mathematics at Cambridge, taking a large cut in income to do mathematics full time. He published over 900 papers and notes in his life.
What did Cayley's 1858 memoir do?
The word "matrix" was introduced by his friend and fellow lawyer James Joseph Sylvester in 1850. Cayley's A Memoir on the Theory of Matrices, published in the Philosophical Transactions of the Royal Society in 1858, made matrices objects you can calculate with. MacTutor's history of matrices and determinants says it contains the first abstract definition of a matrix and gives rules for addition, multiplication, multiplying by a number, and inverses, with the inverse built from the determinant.
In the memoir, reprinted in his Collected Mathematical Papers, Cayley explains that the idea "arises naturally from an abbreviated notation for a set of linear equations". That is exactly how the NCERT Class 12 Matrices chapter uses matrices when solving simultaneous equations.
He also showed that a 2 x 2 matrix satisfies its own characteristic equation, and checked the 3 x 3 case. Then he wrote: "I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree." The general proof came later, from Frobenius. Today it is called the Cayley-Hamilton theorem.
Worked example 1: order matters
Write a matrix by its rows. Let A have rows (1, 1) and (0, 1), and B have rows (1, 0) and (1, 1).
AB: first row (1×1 + 1×1, 1×0 + 1×1) = (2, 1); second row (0×1 + 1×1, 0×0 + 1×1) = (1, 1).
BA: first row (1×1 + 0×0, 1×1 + 0×1) = (1, 1); second row (1×1 + 1×0, 1×1 + 1×1) = (1, 2).
AB has rows (2, 1), (1, 1). BA has rows (1, 1), (1, 2). They are different, so matrix multiplication is not commutative.
Worked example 2: a matrix that satisfies its own equation
Let A have rows (2, 1) and (1, 3). For a 2 x 2 matrix, Cayley's result says A² − (trace A)A + (det A)I = O, where the trace is the sum of the diagonal entries.
Trace = 2 + 3 = 5. Determinant = 2×3 − 1×1 = 5.
A² has rows (2×2 + 1×1, 2×1 + 1×3) = (5, 5) and (1×2 + 3×1, 1×1 + 3×3) = (5, 10).
5A has rows (10, 5) and (5, 15). 5I has rows (5, 0) and (0, 5).
A² − 5A + 5I has rows (5 − 10 + 5, 5 − 5 + 0) = (0, 0) and (5 − 5 + 0, 10 − 15 + 5) = (0, 0).
It works. The theorem is not named in the NCERT chapter, but you can check it with the multiplication you learn there.
Worked example 3: an inverse without the adjoint
From A² − 5A + 5I = O, write A(A − 5I) = −5I, so A⁻¹ = (5I − A)/5. That gives A⁻¹ with rows (3, −1) and (−1, 2), divided by 5. Check: A times the matrix with rows (3, −1), (−1, 2) gives rows (6 − 1, −2 + 2) = (5, 0) and (3 − 3, −1 + 6) = (0, 5), which is 5I. Correct.
Common mistakes with matrix algebra
Expanding (A + B)² as A² + 2AB + B². It is A² + AB + BA + B², and AB and BA are usually different.
Cancelling. AB = AC does not mean B = C unless A has an inverse.
Multiplying entry by entry. Each entry of AB is a row of A times a column of B, added up.
Forgetting order in inverses. (AB)⁻¹ = B⁻¹A⁻¹, not A⁻¹B⁻¹.
Our post on matrices and determinants for JEE has more practice on these. When a step in your working does not match the answer key, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can see where your method went off.
What can a student take from Cayley's way of working?
Use whatever time you have. He wrote about 250 papers while working full time in another job. Short, regular sessions add up.
Name the pattern. Others had used arrays of numbers. Cayley asked what rules those arrays obey on their own, and that question created a subject.
Check small cases honestly. He proved the 2 x 2 case, checked 3 x 3, and said plainly that he had not proved the rest. Say what you have shown, and no more.
Pick any 2 x 2 matrix from your textbook exercise and verify A² − (trace A)A + (det A)I = O for it today.
