Matrices & Determinants — NDA Mathematics

NDA does not ask you to understand matrices at any depth — it asks whether you can run a fixed sequence of mechanical rules without a sign slip: expand a $3\times 3$ determinant, apply a row property, divide by $|A|$ to invert, or set up Cramer's rule. There is no derivation, no proof, no calculator. The whole chapter is procedure executed at speed, and with $-0.8333$ for every wrong answer, a single dropped sign turns a 30-second gain into a net loss.

1. What NDA actually asks

Weightage: ~10% of the Mathematics paper — roughly 10–12 of the 120 questions (25–30 of 300 marks). The syllabus is Class 11–12 CBSE level, and the recurring question families are:

1. Types & order — identify square, diagonal, scalar, identity, symmetric, skew-symmetric matrices; find an unknown entry using a matrix-equality condition.
2. Matrix algebra — addition, scalar multiplication, and (most often) matrix multiplication: computing a product, checking whether a product is even defined, or exploiting $AB\ne BA$.
3. Transpose properties — especially the reversal law $(AB{)}^T=B^T{A}^T$, and symmetric/skew-symmetric decomposition.
4. Determinant evaluation — direct computation of $2\times 2$ and $3\times 3$ determinants by cofactor expansion.
5. Determinant properties — $|kA|=k^n|A|$, $|AB|=|A||B|$, row/column operations, triangular-matrix determinants, and the "determinant of an odd-order skew-symmetric matrix is zero" fact.
6. Adjoint and inverse — computing $A^{-1}$ for a $2\times 2$ (occasionally $3\times 3$) matrix, and properties like $|\text{adj }A|=|A{|}^{n-1}$.
7. Linear systems — Cramer's rule or the matrix-inverse method for 2 or 3 equations, and reading off consistency (unique / no solution / infinite solutions) from $\Delta$.

Nothing here goes beyond board-level content, but NDA compresses it into single-step MCQs that punish slow, careless arithmetic more than they reward theory.

2. Matrix operations & properties

Order and types. A matrix of order $m\times n$ has $m$ rows and $n$ columns. The types that actually get tested:

Occasionally tested extras: orthogonal ($A{A}^T=I$), idempotent ($A^2=A$), involutory ($A^2=I$). Two matrices are equal only if they have the same order and every corresponding entry matches — NDA loves hiding a "find $x,y$" question inside a matrix-equality statement.

Addition and scalar multiplication. Only matrices of the same order can be added — add entry by entry. Addition is commutative and associative. Scalar multiplication $kA$ multiplies every entry by $k$.

Multiplication — the one that trips people up. For $A_{m\times n}\cdot B_{n\times p}=C_{m\times p}$, the inner dimensions must match ($n=n$); the result takes the outer dimensions ($m\times p$). Entry $c_{ij}$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

$C_{ij}={\sum }_{k=1}^n{A}_{ik}{B}_{kj}$

Critical properties: multiplication is associative ($(AB)C=A(BC)$) and distributive over addition, but it is not commutative — $AB\ne BA$ in general, and one may even be undefined while the other is defined. There is also no cancellation law: $AB=O$ does not force $A=O$ or $B=O$.

Transpose. $A^T$ swaps rows and columns. Key rules:

$(A^T{)}^T=A(A+B{)}^T=A^T+B^T(kA{)}^T=k{A}^T(AB{)}^T=B^T{A}^T$

The last one reverses order — the single most-tested transpose fact. Every square matrix splits uniquely into a symmetric and a skew-symmetric part:

$A=\underset{\text{symmetric}}{\underbrace{\frac{1}{2}(A+A^T)}}+\underset{\text{skew-symmetric}}{\underbrace{\frac{1}{2}(A-A^T)}}$

3. Determinants & their properties

A determinant is defined only for a square matrix and reduces it to a single number.

$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

For $3\times 3$, expand along any row or column using minors ($M_{ij}$ = determinant left after deleting row $i$ and column $j$) and cofactors $C_{ij}=(-1{)}^{i+j}{M}_{ij}$, with the sign checkerboard

$\left|\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right|$

Expanding along the first row of $A=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$:

$|A|=a_1(b_2{c}_3-b_3{c}_2)-b_1(a_2{c}_3-a_3{c}_2)+c_1(a_2{b}_3-a_3{b}_2)$

You may expand along any row or column — pick the one with the most zeros to minimise work; the answer is identical.

Properties worth memorising cold:

• Two identical (or proportional) rows/columns $\Rightarrow |A|=0$.
• Interchanging any two rows or columns flips the sign of $|A|$.
• Multiplying one row/column by $k$ multiplies $|A|$ by $k$; scaling the whole $n\times n$ matrix gives $|kA|=k^n|A|$.
• $R_i\to R_i+k{R}_j$ (adding a multiple of one row to another) leaves $|A|$ unchanged — the main tool for creating zeros before expanding.
• $|A^T|=|A|$.
• $|AB|=|A||B|$.
• For a triangular (or diagonal) matrix, $|A|$ = product of the diagonal entries.
• $|A^{-1}|=\frac{1}{|A|}$ (when $A^{-1}$ exists).
• The determinant of a skew-symmetric matrix of odd order is always $0$: since $|A|=|A^T|=|-A|=(-1{)}^n|A|$, an odd $n$ forces $|A|=-|A|$, i.e. $|A|=0$. (This fails for even $n$ — a $2\times 2$ skew-symmetric matrix can have a non-zero determinant.)

4. Inverse of a matrix

A square matrix $A$ is non-singular (invertible) iff $|A|\ne 0$; if $|A|=0$ it is singular and has no inverse.

Adjoint. Build the cofactor matrix (every entry replaced by its cofactor $C_{ij}$), then transpose it:

$\text{adj}(A)=[C_{ij}{]}^T$

The defining identity is $A\cdot \text{adj}(A)=\text{adj}(A)\cdot A=|A|I$, which gives the inverse formula:

$A^{-1}=\frac{1}{|A|}\text{adj}(A),|A|\ne 0$

Fast $2\times 2$ shortcut — swap the diagonal entries, negate the off-diagonal entries, divide by the determinant:

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\Rightarrow \text{adj}(A)=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\Rightarrow A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Useful corollaries:

• $|\text{adj}(A)|=|A{|}^{n-1}$ for an $n\times n$ matrix — for $3\times 3$, that's $|A{|}^2$.
• $(A^{-1}{)}^{-1}=A$.
• $(AB{)}^{-1}=B^{-1}{A}^{-1}$ — order reverses, exactly like transpose.
• Never forget the $\frac{1}{|A|}$ factor — writing $\text{adj}(A)$ alone as "the inverse" is the single most common NDA slip on this topic.

5. Solving linear systems

Write the system as $AX=B$, where $A$ is the coefficient matrix, $X$ the column of unknowns, $B$ the column of constants.

Matrix (inverse) method. If $|A|\ne 0$: $X=A^{-1}B$ — a unique solution exists.

Cramer's rule (equivalent, often faster for MCQs). For $a_{1}x+b_{1}y+c_{1}z=d_1,a_{2}x+b_{2}y+c_{2}z=d_2,a_{3}x+b_{3}y+c_{3}z=d_3$ let $\Delta$ be the determinant of the coefficient matrix, and let ${\Delta }_x,{\Delta }_y,{\Delta }_z$ be $\Delta$ with the $x$-, $y$-, $z$-coefficient column replaced by the constants column $(d_1,d_2,d_3)$. Then, when $\Delta \ne 0$:

$x=\frac{{\Delta }_x}{\Delta },y=\frac{{\Delta }_y}{\Delta },z=\frac{{\Delta }_z}{\Delta }$

Consistency conditions — the part NDA loves to test conceptually, not just computationally:

Homogeneous systems ($AX=O$, i.e. all constants are $0$) always have the trivial solution $x=y=z=0$. A non-trivial solution exists if and only if $|A|=0$.

7. Common traps

• Sign errors in cofactor expansion. The checkerboard is $+,-,+/-,+,-/+,-,+$. Losing a minus sign on the middle term is the single most frequent error in $3\times 3$ determinants — write the signs down before you compute anything.
• Forgetting the $\frac{1}{|A|}$ in the inverse. $\text{adj}(A)$ is not $A^{-1}$; it is $|A|$ times too large. Always divide.
• Assuming $AB=BA$. Matrix multiplication is not commutative — never reorder a product to "simplify" it.
• Multiplying matrices with mismatched inner dimensions. Write the orders down first: $A_{m\times n}\cdot B_{p\times q}$ is defined only if $n=p$, and the result is $m\times q$ — not $n\times q$ or $m\times p$.
• Writing $(AB{)}^T=A^T{B}^T$. The order reverses: $(AB{)}^T=B^T{A}^T$ (same for inverses: $(AB{)}^{-1}=B^{-1}{A}^{-1}$).
• Treating $\Delta =0$ as automatically "infinite solutions". It can equally mean no solution — check ${\Delta }_x,{\Delta }_y,{\Delta }_z$ before concluding either way.
• Applying $|kA|=k|A|$ instead of $k^n|A|$. Every one of the $n$ rows picks up a factor of $k$, so a $3\times 3$ matrix scales its determinant by $k^3$, not $k$.
• Assuming every skew-symmetric matrix has determinant $0$. True only for odd order; a $2\times 2$ skew-symmetric matrix generally does not.

Run through this list once before the exam and once more with 10 minutes left on the clock — nearly every wrong answer in this chapter traces back to one line on it.