Higher Mathematics 1st Paper · Classes 11-12 · Chapter 1 · 1.3-1.4

Determinants and Finding Their Value.

Two grids, the same numbers: one in [ ], one between | |. The first is an arrangement of numbers; the second is a single number. In this note we pull that one number out of the grid between the bars, in three ways.

1. A determinant is a function

The rule that gives one number for every square matrix is called the determinant. So it is a function: each element of its domain is a square matrix, and each element of its range is a real or complex number. The determinant of a square matrix A = [a_ij] of order n is written det(A) or |A|. Expand it and it becomes one definite number.

A matrix and a determinant are not the same thing. A matrix is a rectangular array of numbers, of any order m × n, written in [ ] or ( ). A determinant exists only for a square matrix, is written | | or det, and its value is a single number.

2. First order and second order

The first order is the easiest: one row, one column, and the value is that number itself. |a₁₁| = a₁₁. So |5| = 5 and |−6| = −6. Here | | does not mean absolute value; it means determinant.

The second order needs two products: the product along ↘ minus the product along ↙.

det[[a₁₁, a₁₂], [a₂₁, a₂₂]] = a₁₁a₂₂ − a₁₂a₂₁

Example: det[[7, −2], [8, −2]] = 7(−2) − (−2)(8) = −14 + 16 = 2.

Change the entries: the ↘ product minus the ↙ product. On the right, the two columns span a parallelogram.

3. The Sarrus diagram

For the third order, the book's first method is the Sarrus diagram. Write the columns side by side twice, then put + before each product along ↘ and − before each product along ↙, and add:

|A| = a₁₁a₂₂a₃₃ + a₁₂a₂₃a₃₁ + a₁₃a₂₁a₃₂ − a₁₃a₂₂a₃₁ − a₁₁a₂₃a₃₂ − a₁₂a₂₁a₃₃

Six lines at once would crowd the grid, so add them one at a time. The three plus lines are blue, the three minus lines red. At the end, type your own value in the box. It goes nowhere; the notebook is yours.

Apply Sarrus to the book's task and det[[1, 4, 0], [−1, 2, 2], [0, 0, 2]] = 12 and det[[p, q, r], [0, x, y], [0, 0, z]] = pxz. In the second, everything below the principal diagonal is zero, so only one of the six products survives. That is the rule: the determinant of a triangular (upper or lower) or diagonal matrix is the product of the principal diagonal entries. det[[2, 4, 6], [0, 8, 10], [0, 0, 12]] = 2 × 8 × 12 = 192.

Factorials can be entries too. In det[[2!, 3!, 2!], [3!, 2!, 4!], [0!, 3!, 2!]] you must take 0! = 1, not zero. The entries are then 2, 6, 2; 6, 2, 24; 1, 6, 2 and the value is 2(4 − 144) − 6(12 − 24) + 2(36 − 2) = −280 + 72 + 68 = −140.

4. Expanding along a row or column

The second method is expansion. Along the first row: the (1, 1)-th entry × its remaining determinant, minus the (1, 2)-th entry × its remaining determinant, plus the (1, 3)-th entry × its remaining determinant.

|A| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)

Example: det[[2, 1, 1], [0, 5, −2], [1, −3, 4]] = 2·det[[5, −2], [−3, 4]] − 1·det[[0, −2], [1, 4]] + 1·det[[0, 5], [1, −3]] = 2(20 − 6) − (0 + 2) + (0 − 5) = 28 − 2 − 5 = 21. It agrees with the Sarrus 21.

The first row is not compulsory. Expand along any row or any column and the value is the same; only the sign of the (p, q)-th entry must be (−1)^p+q. The signs alternate like a chessboard: + − +, − + −, + − +. Tap the five chips and compare.

The same determinant gave 21 by four routes, so the choice of route is yours. Take the row or column with the most zeros and the work shrinks. Expanding det[[0, 3, 2x + 7], [2, 7x, 9 + 5x], [0, 0, 2x + 5]] along the 1st column gives −2·3(2x + 5) = −6(2x + 5), which is 0 when x = −5/2.

5. Finding the unknown when the determinant is zero

Board questions often hide an x or a p inside the determinant and say it equals zero. The job has three steps: expand, simplify to an equation, then solve for the unknown. Think first, type your answer in the box, then pick “see the working” to compare.

6. Check yourself

The first question, in the exam's style.

The second comes from a slightly different side.

In the last one, watch the signs.

The whole note on seven cards.

Source: Higher Mathematics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Matrices and Determinants”, section 1.3-1.4.