The Number That Decides

A determinant assigns a single number to a square matrix; it is written |A|, and it is worth stressing upfront that this notation is read as 'the determinant of A', a completely different idea from the modulus (absolute value) of a number, even though both use vertical bars. Only square matrices have determinants at all; a rectangular matrix simply has none. For a 1x1 matrix, the determinant is just its single entry. For a 2x2 matrix [[a,b],[c,d]], the determinant is ad-bc. Checking |[[6,2],[5,3]]|: (6)(3)-(2)(5) = 18-10 = 8.

For a 3x3 matrix, the determinant is found by expanding along any row or column, using each entry multiplied by its own minor (the smaller determinant left after deleting that entry's row and column), with alternating signs. Take A = [[2,0,1],[3,1,4],[1,0,2]]. Expanding along row 1: 2 times (1x2-4x0), minus 0 times (its own minor), plus 1 times (3x0-1x1), which works out to 2(2) - 0 + 1(-1) = 4-1 = 3. Expanding instead along column 2, which conveniently has two zeros (positions (1,2) and (3,2)): only the middle entry, 1, contributes, giving 1 times the minor left by deleting row 2 and column 2, which is det[[2,1],[1,2]] = 4-1 = 3. Both expansions, along a full row and along a mostly-zero column, agree exactly: |A| = 3.

If every entry of a matrix is scaled by a constant k, the determinant scales by k raised to the power of the matrix's order, not just by k itself. Take A = [[1,3],[2,4]], with |A| = (1)(4)-(3)(2) = 4-6 = -2. Now let B = 2A = [[2,6],[4,8]], so |B| = (2)(8)-(6)(4) = 16-24 = -8. Checking against the rule |kA| = k^n |A| with k=2 and n=2 (since A is 2x2): 2^2 times -2 = 4 times -2 = -8, matching |B| exactly. Scaling every entry of a 2x2 matrix by 2 does not merely double the determinant, it quadruples it, because BOTH the row and column directions get scaled at once.

Continuing with A = [[2,0,1],[3,1,4],[1,0,2]] from earlier, the cofactors of row 1 are: A11 = +(1x2-4x0) = 2, A12 = -(3x2-4x1) = -2, and A13 = +(3x0-1x1) = -1. Multiplying row 1's own entries by these cofactors: (2)(2) + (0)(-2) + (1)(-1) = 4+0-1 = 3, exactly matching |A| found earlier, confirming a row's entries times its OWN cofactors reproduce the determinant. But multiplying row 2's entries by row 1's cofactors instead: (3)(2) + (1)(-2) + (4)(-1) = 6-2-4 = 0. This is not a coincidence: using a row's entries with a DIFFERENT row's cofactors always gives exactly zero, a fact that turns out to underpin why the adjoint formula for the inverse works at all.

The area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is half the absolute value of a 3x3 determinant built from the coordinates plus a column of 1's. For a right triangle with vertices (0,0), (5,0) and (0,6): the determinant det[[0,0,1],[5,0,1],[0,6,1]] works out to 0-0+1(30-0) = 30, giving an area of (1/2)(30) = 15 square units, matching the elementary formula (1/2)(base)(height) = (1/2)(5)(6) = 15 exactly. This same determinant, when it comes out to exactly ZERO, signals something else entirely: the three points do not form a real triangle at all, because they lie on one straight line. Checking (1,1), (2,3) and (4,7): det[[1,1,1],[2,3,1],[4,7,1]] = 1(3-7) - 1(2-4) + 1(14-12) = -4+2+2 = 0, confirming these three points are collinear.

The adjoint of a matrix, adj(A), is the TRANSPOSE of its cofactor matrix. For a 2x2 matrix [[a,b],[c,d]], there is a shortcut: adj(A) = [[d,-b],[-c,a]], swap the diagonal entries, negate the off-diagonal ones. Take A = [[5,2],[3,1]]: |A| = (5)(1)-(2)(3) = 5-6 = -1, and adj(A) = [[1,-2],[-3,5]]. A central theorem states A(adj A) = |A| times the identity matrix; checking: A times adj(A) gives [[-1,0],[0,-1]], which is exactly -1 times the identity, matching |A| = -1. This theorem, once |A| is not zero, can be rearranged directly into the long-awaited formula: A^-1 = adj(A) / |A|. For this A, A^-1 = [[1,-2],[-3,5]] divided by -1, giving [[-1,2],[3,-5]]. Checking A times A^-1: [[5,2],[3,1]] times [[-1,2],[3,-5]] gives [[1,0],[0,1]], confirming the inverse is correct. This resolves exactly the cliffhanger left by the previous chapter: a matrix is invertible precisely when it is non-singular, meaning |A| is not zero, and the adjoint hands over the actual formula.

A system like x+y=15 and 3x-y=5 can be written as AX=B, with A=[[1,1],[3,-1]], X the column of unknowns, and B=[[15],[5]]. Since |A| = (1)(-1)-(1)(3) = -4, A is non-singular, so X = A^-1 B gives the unique solution directly: computing adj(A) = [[-1,-1],[-3,1]] and A^-1 = adj(A)/(-4), then multiplying by B gives x=5 and y=10. Checking: 5+10=15 and 3(5)-10=5, both correct. This same matrix method, one clean multiplication once A^-1 is known, is exactly the modern descendant of a very old idea: ancient Chinese counting-rod methods for eliminating unknowns across several equations, the 1683 work of the Japanese mathematician Seki Kowa on early determinant-like expansions, and the term 'determinant' itself, coined in its modern sense by Cauchy in 1812. From letter-numbers standing for a single unknown in Class 7, all the way to one matrix equation solving several unknowns at once here, the whole thread has been the same idea, growing up.

The matrix method X = A^-1 B only works when A is non-singular, but a system of equations is still worth examining even when |A| = 0. A system is called consistent if it has at least one solution, and inconsistent if it has none. When |A| is not zero, the system is always consistent with exactly one unique solution. When |A| equals zero, a further check decides the outcome: compute (adj A) times B. If this product is NOT the zero matrix, the system is inconsistent (no solution exists at all). If (adj A) times B IS the zero matrix, the system may have infinitely many solutions or may still have none, a case needing methods beyond this chapter to fully resolve. This three-way split, unique solution, no solution, or a case needing further work, is the complete honest picture of what a determinant of zero can mean for a system of equations.