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Matrices and determinants

Further Mathematics · WAEC and JAMB · SS2 and SS3

Matrices give easy marks for multiplication and inverses, then test you on solving simultaneous equations. The examiner is checking whether you can keep rows and columns straight under pressure, which is where most of the lost marks go.

What you need to know

  • The order of a matrix is rows by columns, always in that order, so a 2 by 3 matrix has 2 rows and 3 columns. Two matrices can be added only when they have exactly the same order, and you add corresponding entries.
  • Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B. The product then has the number of rows of A and the number of columns of B, so a 2 by 3 times a 3 by 2 gives a 2 by 2.
  • To find an entry of a product, run along the row of the first matrix and down the column of the second, multiplying pairs and adding. Row one times column two gives the entry in row one, column two.
  • Matrix multiplication is not commutative: AB and BA are usually different, and may not both exist. Never cancel matrices as you would numbers, and never write A/B, because matrix division does not exist.
  • For a 2 by 2 matrix with rows (a, b) and (c, d), the determinant is ad - bc. This single number controls everything: if it is zero the matrix is singular, has no inverse, and the related equations have either no solution or infinitely many.
  • The inverse of that 2 by 2 matrix is 1/(ad - bc) times the matrix with rows (d, -b) and (-c, a). Swap the leading diagonal entries, change the sign of the other two, then divide by the determinant.
  • Check an inverse by multiplying: A times A inverse must give the identity matrix with rows (1, 0) and (0, 1). This takes a few seconds and catches a wrong sign immediately.
  • A 3 by 3 determinant is expanded along the first row with alternating signs plus, minus, plus. Each entry multiplies the 2 by 2 determinant left when you delete its own row and column, and that 2 by 2 determinant is called its minor.
  • The sign pattern for a 3 by 3 matrix is plus, minus, plus across the first row, minus, plus, minus across the second. Forgetting the minus on the middle term of the first row is the single commonest 3 by 3 error.
  • You may expand a determinant along any row or column, so choose the one containing the most zeros. An entry of zero kills its whole minor, which saves you an entire 2 by 2 calculation.
  • To solve simultaneous equations by matrix methods, write them as AX = B, then X = A inverse times B. Order matters: the inverse goes in front of B, not behind it.
  • Cramer's rule solves the same system by determinants: x = det(A_x)/det(A), where A_x is A with its first column replaced by B, and similarly for y. Both methods are accepted, so use whichever you can execute cleanly.
  • The transpose of a matrix swaps rows and columns. A matrix equal to its own transpose is symmetric, and det(A) = det(A transpose), which is why expanding along a column is as valid as expanding along a row.

Key terms

Matrix
A rectangular array of numbers arranged in rows and columns and treated as a single object.
Order of a matrix
Its size, stated as number of rows by number of columns.
Determinant
A single number computed from a square matrix which is zero exactly when the matrix has no inverse.
Singular matrix
A square matrix whose determinant is zero, and which therefore has no inverse.
Identity matrix
The square matrix with ones along the leading diagonal and zeros elsewhere, which leaves any matrix unchanged under multiplication.
Minor
The determinant of the smaller matrix left after deleting the row and the column of a chosen entry.
Transpose
The matrix obtained by writing the rows of a given matrix as its columns.

Formulae

  • For A with rows (a, b) and (c, d): det(A) = ad - bc
  • A inverse = (1/(ad - bc)) times matrix with rows (d, -b) and (-c, a)
  • A times A inverse = I, the identity matrix
  • 3 by 3 determinant along the first row: a(ei - fh) - b(di - fg) + c(dh - eg)
  • Sign pattern for cofactors: plus minus plus / minus plus minus / plus minus plus
  • AX = B gives X = A inverse times B
  • Cramer's rule: x = det(A_x)/det(A) and y = det(A_y)/det(A)
  • det(AB) = det(A) x det(B)
  • det(A transpose) = det(A)
  • A matrix is singular when det(A) = 0

Worked examples

Use the inverse matrix method to solve the simultaneous equations 2x + 3y = 13 and x + 4y = 14.

  1. Write as AX = B where A has rows (2, 3) and (1, 4), X has entries x and y, and B has entries 13 and 14.
  2. Find the determinant: det(A) = (2)(4) - (3)(1) = 8 - 3 = 5. It is not zero, so an inverse exists.
  3. Form the inverse: A inverse = (1/5) times the matrix with rows (4, -3) and (-1, 2).
  4. Multiply: x = (1/5)[(4)(13) + (-3)(14)] = (1/5)(52 - 42) = 10/5 = 2.
  5. And y = (1/5)[(-1)(13) + (2)(14)] = (1/5)(-13 + 28) = 15/5 = 3.
  6. Check in both original equations: 2(2) + 3(3) = 4 + 9 = 13, and 2 + 4(3) = 2 + 12 = 14.

Evaluate the determinant of the matrix M with rows (1, 2, 3), (0, 4, 5) and (1, 0, 6).

  1. Expand along the first row using the signs plus, minus, plus.
  2. First term: 1 times the determinant of rows (4, 5) and (0, 6), which is 1[(4)(6) - (5)(0)] = 1(24) = 24.
  3. Second term: minus 2 times the determinant of rows (0, 5) and (1, 6), which is -2[(0)(6) - (5)(1)] = -2(-5) = +10.
  4. Third term: plus 3 times the determinant of rows (0, 4) and (1, 0), which is 3[(0)(0) - (4)(1)] = 3(-4) = -12.
  5. Add the three results: 24 + 10 - 12 = 22.
  6. Confirm by expanding along the first column instead: 1(24 - 0) - 0 + 1[(2)(5) - (3)(4)] = 24 + (10 - 12) = 24 - 2 = 22.

The mistake to avoid

Dropping the minus sign on the middle term of a 3 by 3 expansion, which turns 24 + 10 - 12 into 24 - 10 - 12 and loses the question. In the 2 by 2 inverse, the other standing error is swapping b and c instead of swapping a and d; it is the leading diagonal entries that change places, while b and c only change sign.

In the exam

Compute the determinant first in every inverse question and write it down on its own line, because if it is zero you must say the matrix is singular rather than grinding out an impossible inverse. Multiply your inverse by the original matrix to confirm you get the identity before you use it, and always show the matrix equation AX = B explicitly, since that structure alone earns a mark.