Matrices and Determinants

Lakshya NDA

Master matrix operations, determinant properties and linear equation solving.

Level

Core

Study Time

20 min

Exam Relevance

High (formula + property-driven questions)

Core concepts

Matrix types and operations.
Determinant expansion and properties.
Adjoint and inverse of square matrix.
Solving linear equations with matrix method and Cramer's rule.

Must-know shortcuts

If two rows/columns are same -> determinant is 0.
Interchanging two rows changes sign.
Factor common constants before expansion.

Question style in NDA

Value of determinant after row/column transformation.
Inverse existence check (|A| != 0).
Solving 2-variable or 3-variable system quickly.

Common mistakes

Expanding along a row with many non-zero terms (wastes time).
Sign errors in cofactors.
Forgetting determinant condition before inverse.

Chapter-end mini quiz

5 quick questions from the question bank.

Question 1

If $\begin{vmatrix} a & -b & a-b-c \\ -a & b & -a+b-c \\ -a & -b & -a-b+c \end{vmatrix} - kabc = 0$ where $(a \neq 0,\ b \neq 0,\ c \neq 0)$, then what is the value of $k$?

$-4$$-2$$2$$4$

Question 2

What is $A^{-1}$ equal to?

$\begin{bmatrix} 1 & -1 & 0 \\ -2 & 3 & -4 \\ -2 & 3 & -3 \end{bmatrix}$$\begin{bmatrix} 1/2 & -1/2 & 0 \\ -1 & 3/2 & -2 \\ -1 & 3/2 & -3/2 \end{bmatrix}$$\begin{bmatrix} 2 & -2 & 0 \\ -4 & 6 & -8 \\ -4 & 6 & -6 \end{bmatrix}$$\begin{bmatrix} 1/5 & -1/5 & 0 \\ -2/5 & 3/5 & -4/5 \\ -2/5 & 3/5 & -3/5 \end{bmatrix}$

Question 3

If $a, b, c$ are the sides of triangle $ABC$, then what is the determinant $\begin{vmatrix} a^2 & b\sin A & c\sin A \\ b\sin A & 1 & \cos A \\ c\sin A & \cos A & 1 \end{vmatrix}$ equal to?

ZeroArea of trianglePerimeter of triangle$a^2 + b^2 + c^2$

Question 4

If $M_k = \begin{bmatrix} k & k-1 \\ k-1 & k \end{bmatrix}$ where $k$ is a natural number, then what is $|M_1| + |M_2| + |M_3| + \ldots + |M_{50}|$ equal to?

50125025005000

Question 5

If $D_n = \begin{vmatrix} n & 20 & 30 \\ n^2 & 40 & 50 \\ n^3 & 60 & 70 \end{vmatrix}$, then what is the value of $\sum_{n=1}^{4} D_n$?

$-10000$$-10$$10$$10000$

Source Base

NCERT Class XII Mathematics: Matrices and Determinants
CBSE board objective sets on determinant properties

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