Split Down the Middle

A small business tracking two products (notebooks and pens) sold across three months can lay its sales out as a rectangular table: 3 rows for the months, 2 columns for the products. This rectangular arrangement of numbers is called a matrix, and its order describes its shape directly: a matrix with m rows and n columns is an m x n matrix. Each individual entry is labelled a_ij, where i is its row number and j its column number, so a_23 means the entry sitting in row 2, column 3. A single row of numbers is a row matrix; a single column is a column matrix; a matrix with equal rows and columns (m=n) is a square matrix.

Beyond the basic row/column/square distinction, a few special matrix types come up constantly. A diagonal matrix is a square matrix where every entry OFF the main diagonal is zero. A scalar matrix is a diagonal matrix where every diagonal entry is additionally equal to the same constant k. The identity matrix, written I, is the special scalar matrix where that constant k is exactly 1 -- every identity matrix is a scalar matrix, but not every scalar matrix is an identity matrix. A zero (or null) matrix, written O, has every single entry equal to 0, regardless of its shape. Two matrices are called equal only when they share the exact same order AND every corresponding entry matches -- both conditions are required, not just one.

Matrix addition is only defined between two matrices of the EXACT same order, done entry by entry: adding the corresponding entries of each matrix together. Scalar multiplication instead takes one matrix and one plain number k, multiplying every single entry of the matrix by k. Subtraction combines both ideas: A - B is defined as A + (-1)B, scaling B by -1 and then adding. These operations obey familiar-looking rules -- addition is commutative and associative, has the zero matrix as an identity, and every matrix has an additive inverse (-A) -- all provable directly from the fact that ordinary number addition, applied entry by entry, already has these properties.

Matrix multiplication is genuinely different from the entrywise operations above. The product AB is only defined when A's number of COLUMNS matches B's number of ROWS, and each entry of the result is found by pairing up a row of A with a column of B, multiplying matching positions together, and summing: for A = [[1,2],[3,4]] and B = [[5,6],[7,8]], the top-left entry of AB is (1)(5)+(2)(7) = 5+14 = 19, and working through all four entries gives AB = [[19,22],[43,50]]. Multiplication is NOT commutative in general -- computing BA for the same two matrices gives [[23,34],[31,46]], a genuinely different result from AB. Even more surprisingly, matrices can multiply to give the zero matrix without either factor being zero itself: A = [[1,2],[3,6]] (row 2 is exactly 3 times row 1) and B = [[-2,-4],[1,2]] (column 2 is exactly 2 times column 1) are both nonzero, yet every entry of AB works out to exactly 0 -- a genuine departure from ordinary number arithmetic, where xy=0 always forces x=0 or y=0.

The transpose of a matrix A, written A' (or A^T), is formed by turning every row of A into the matching column: if A is m x n, its transpose A' is n x m. Four properties always hold: transposing twice returns the original matrix, (A')' = A; a scalar factors out unchanged, (kA)' = kA'; transposing a sum equals the sum of the transposes, (A+B)' = A'+B'; and transposing a PRODUCT reverses the order, (AB)' = B'A', not A'B'. This last property, order-reversal under a combined operation, is a pattern worth remembering, since it reappears identically for matrix inverses later in this same chapter.

A square matrix is symmetric if it equals its own transpose: A' = A. For A = [[4,7,-2],[7,9,5],[-2,5,1]], swapping rows and columns gives back the exact same matrix, since the entry above the diagonal always matches the entry below it in the mirrored position (7 and 7, -2 and -2, 5 and 5). A square matrix is skew-symmetric if it equals the NEGATIVE of its own transpose: A' = -A. For B = [[0,3,-6],[-3,0,8],[6,-8,0]], every mirrored pair has flipped signs (3 and -3, -6 and 6, 8 and -8) instead of matching exactly.

This is not a coincidence spotted by example; it follows directly from the definition. If A' = -A, then look specifically at a diagonal entry a_ii. Transposing a matrix never moves a diagonal entry (row i, column i stays row i, column i), so the (i,i) entry of A' is just a_ii itself. But A' = -A means the (i,i) entry of A' must ALSO equal -a_ii. Putting these together: a_ii = -a_ii, which forces 2 times a_ii to equal 0, so a_ii = 0. This holds for every diagonal position at once, proving every skew-symmetric matrix, of any size, must have an entirely zero diagonal.

Take A = [[3,7],[1,5]], which is neither symmetric nor skew-symmetric (7 does not equal 1, and 1 does not equal -7). Its transpose is A' = [[3,1],[7,5]]. Now compute (1/2)(A+A'): A+A' = [[6,8],[8,10]], so half of that is [[3,4],[4,5]], which IS symmetric (4 matches 4). Separately, (1/2)(A-A'): A-A' = [[0,6],[-6,0]], so half of that is [[0,3],[-3,0]], which IS skew-symmetric (zero diagonal, 3 and -3 flip sign). Adding the two halves back together: [[3,4],[4,5]] + [[0,3],[-3,0]] = [[3,7],[1,5]], exactly recovering A. This is not a coincidence for this one matrix: for ANY square matrix A, (1/2)(A+A') is always symmetric and (1/2)(A-A') is always skew-symmetric, and the two halves always add back to A exactly.

With ordinary numbers, if xy=0, then x=0 or y=0, no exceptions. Matrices break this rule completely. Let A = [[1,2],[3,6]] (notice row 2 is exactly 3 times row 1) and B = [[-2,-4],[1,2]] (notice column 2 is exactly 2 times column 1). Multiplying: the top-left entry of AB is (1)(-2)+(2)(1) = -2+2 = 0; the top-right is (1)(-4)+(2)(2) = -4+4 = 0; the bottom-left is (3)(-2)+(6)(1) = -6+6 = 0; the bottom-right is (3)(-4)+(6)(2) = -12+12 = 0. Every entry of AB is zero, so AB is the zero matrix, even though neither A nor B is the zero matrix. Matrices that behave this way are sometimes called zero-divisors, a genuinely surprising departure from ordinary number arithmetic.

A square matrix A is called invertible if there exists another square matrix B, of the same order, such that AB = BA = I (the identity matrix). Checking A = [[3,1],[2,1]] against B = [[1,-1],[-2,3]]: AB gives top-left (3)(1)+(1)(-2)=3-2=1, top-right (3)(-1)+(1)(3)=-3+3=0, bottom-left (2)(1)+(1)(-2)=2-2=0, bottom-right (2)(-1)+(1)(3)=-2+3=1, so AB=I. Checking BA the other way round confirms BA=I too. So A is invertible, and B is written A^-1. A genuinely useful fact: this inverse, if it exists at all, is always unique. If both B and C were inverses of A, then B = BI = B(AC) = (BA)C = IC = C, using nothing but the definition and the associative property of matrix multiplication, proving B and C must be the very same matrix.

If A and B are both invertible matrices of the same order, then AB is also invertible, and (AB)^-1 = B^-1 times A^-1, with the order reversed. This makes sense the same way undoing two combined steps does: if a process is 'do A, then do B', undoing it means 'undo B first, then undo A', the exact reverse order. The same order-reversal pattern shows up for transposes too: (AB)' = B'A', not A'B', a completely different operation obeying the identical reversal rule. Recognising this pattern, that reversing a combined operation always means reversing BOTH the steps AND their order, saves having to re-derive it separately every time it comes up.

Everything covered so far describes what an invertible matrix IS, and proves solid, general facts about inverses once they exist: that they're unique, and that inverting a product reverses the order. But none of it explains how to actually CONSTRUCT an inverse from scratch, starting only from a matrix A and no ready-made candidate B to check. Checking a guessed B against A, as done above, works fine when a candidate is already in hand, but it is not a method for producing one from nothing. That missing piece, a genuine, dependable way to build A^-1 for any invertible A, and a clean test for deciding whether A is invertible at all, needs one more tool entirely: the determinant.