Matrices and determinants for NET: properties, shortcuts and worked MCQs

Matrices and determinants on NET reward properties more than arithmetic. If you know when a determinant is zero without expanding it, how scaling changes it, and how to invert a 2x2 in your head, most questions take under 40 seconds. The rest is Cramer's rule and a few definitions.

Key facts

• Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
• Matrices and Determinants is unit 4 of the Class 11 PECTAA mathematics book (National Curriculum 2023)
• Our chapter map estimates about 7 MCQs from this chapter. NUST does not publish chapter counts

Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map

What the chapter covers#

The chapter covers matrix algebra (addition, multiplication, transpose), determinants, inverses, and solving linear systems with Cramer's rule. It also defines the special matrices: symmetric, skew-symmetric, diagonal, scalar, identity, singular and orthogonal.

NET questions come in three kinds. Definition questions ask which matrix is symmetric or what a cofactor is. Property questions give you |A| and ask for |2A| or |A^-1|. Calculation questions ask for a determinant, an inverse or the solution of a system. The property questions are where prepared students gain the most time.

Formulas and properties#

Algebra:

(AB)^t   = B^t A^t
(AB)^-1  = B^-1 A^-1
AB ≠ BA in general
A^-1     = adj(A) / |A|      exists only if |A| ≠ 0

for A = [a  b]
        [c  d]
|A|  = ad - bc
A^-1 = (1/(ad - bc)) [ d  -b]
                     [-c   a]

Determinant properties, for an n x n matrix A:

|kA|     = k^n |A|
|A^t|    = |A|
|AB|     = |A| |B|
|A^-1|   = 1 / |A|
|adj A|  = |A|^(n-1)

|A| = 0 if two rows (or columns) are equal or proportional
|A| = 0 if a row (or column) is all zeros
swapping two rows changes the sign of |A|
adding a multiple of one row to another leaves |A| unchanged

cofactor  A_ij = (-1)^(i+j) M_ij

Special matrices:

symmetric        A^t = A
skew-symmetric   A^t = -A     (odd order means |A| = 0)
orthogonal       A A^t = I    (so |A| = ±1)
singular         |A| = 0

Cramer's rule for a system AX = B with |A| ≠ 0: replace the column of the unknown with B, take that determinant, divide by |A|.

For any 2x2 matrix, A^2 - (trace A) A + |A| I = 0. This is the Cayley-Hamilton result, and it solves a whole class of NET questions in one line.

Worked MCQs#

Five questions in the bank's style. For each one we did the calculation, then checked it a second way where one exists.

1. A is a 4x4 matrix with |A| = 5. Find |2A|.

|2A| = 2^4 x 5 = 80

The wrong option is 10, from forgetting the power.

2. Evaluate the determinant with rows (42, 1, 6), (28, 7, 4), (14, 3, 2).

Look before you expand. Column 1 is 42, 28, 14. Column 3 is 6, 4, 2. Column 1 is 7 times column 3, so the determinant is 0.

3. If A has rows (1, 0) and (-1, 7), and A^2 = 8A + kI, find k.

Use Cayley-Hamilton. The trace is 1 + 7 = 8 and |A| = 7 - 0 = 7:

A^2 - 8A + 7I = 0
A^2 = 8A - 7I,   so k = -7

Check by multiplying: A^2 has rows (1, 0) and (-8, 49), and 8A - 7I has the same rows.

4. Evaluate the determinant with rows (1, -3, -6), (0, 4, -3i), (0, -3i, 4).

Expand along the first column, which has two zeros:

1 x [4(4) - (-3i)(-3i)] = 16 - 9i^2 = 16 + 9 = 25

5. Solve 2x + y = 5 and x - y = 1 by Cramer's rule.

|A|  = (2)(-1) - (1)(1) = -3
|Ax| = (5)(-1) - (1)(1) = -6      x = -6 / -3 = 2
|Ay| = (2)(1)  - (5)(1) = -3      y = -3 / -3 = 1

Check: 2(2) + 1 = 5 and 2 - 1 = 1.

Common traps#

• |kA| vs k|A|. The scalar comes out once per row, so it is k^n. Example 1 above is built on this trap.
• Order of multiplication. (AB)^t is B^t A^t, not A^t B^t. The same reversal holds for inverses.
• Order of the result. A 2x3 times a 3x4 gives a 2x4. If the inner numbers differ, the product does not exist. Subtraction needs identical orders.
• Inconsistent systems. x + 2y = 11 and -3x - 6y = 33 look solvable, but the second says x + 2y = -11. Parallel lines, no solution.
• i inside a determinant. (-3i)(-3i) = 9i^2 = -9. Students write +9 and get 7 instead of 25.

How to prepare#

Learn the property block first. It is short and it covers more questions than the calculation methods do. Then practise 3x3 expansion along the row or column with the most zeros. Finally, do a timed mixed set.

This chapter links to complex numbers through determinants with i in them, and to vectors in space through the scalar triple product, which is a 3x3 determinant. The NET mathematics guide covers how to order these chapters.

FAQ#