JEE Main Maths Matrices & Determinants 2027: Inverse, Adjoint & Solving Linear Systems

Matrices and determinants form a compact, formula-driven chapter, and that is exactly why it is such a dependable source of marks. A handful of properties turn heavy-looking calculations into two-line answers, and the same tools solve systems of equations without any elimination at all.

Know the properties, and most determinants never need to be expanded fully.

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Chapter at a Glance

Priority reflects past-paper trends, not an official NTA weightage.

What the NTA Syllabus Covers

• Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three
• Evaluation of determinants, area of triangles using determinants
• Adjoint and inverse of a square matrix
• Test of consistency and solution of simultaneous linear equations in two or three variables using matrices

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Master These Topics

1. Determinant Properties That Save Time

For a square matrix $A$ of order $n$:

• $\lvert AB \rvert = \lvert A \rvert \lvert B \rvert$ and $\lvert A^T \rvert = \lvert A \rvert$
• $\lvert kA \rvert = k^n \lvert A \rvert$
• Swapping two rows changes the sign. Two identical rows give a determinant of $0$.
• Adding a multiple of one row to another row leaves the value unchanged.

Worked example (order 3): Expanding along the first row,

$$\begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1$$

Trap: In cofactor expansion the signs alternate in a checkerboard pattern starting with $+$ in the top-left corner.

2. Adjoint and Inverse

The inverse exists only if $\lvert A \rvert \ne 0$, and it is given by $A^{-1} = \dfrac{\operatorname{adj} A}{\lvert A \rvert}$. Useful identities: $A(\operatorname{adj} A) = \lvert A \rvert I$, $\lvert \operatorname{adj} A \rvert = \lvert A \rvert^{n-1}$ and $(AB)^{-1} = B^{-1}A^{-1}$.

Worked example (order 2): For $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$, $\lvert A \rvert = 6 - 5 = 1$. Swap the diagonal entries and change the signs of the off-diagonal ones:

$$A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$$

Trap: $AB \ne BA$ in general, and $(AB)^T = B^T A^T$. The order reverses.

3. Area of a Triangle Using Determinants

The area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is

$$\text{Area} = \frac{1}{2}\left\lvert \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right\rvert$$

For the vertices $(1, 1), (4, 1), (1, 5)$ the determinant is $1(1-5) - 1(4-1) + 1(20-1) = -4 - 3 + 19$... more simply, subtracting rows reduces it to a base of 3 and a height of 4, giving an area of $\tfrac{1}{2} \cdot 3 \cdot 4 = 6$. Three points are collinear exactly when this determinant is zero.

4. Solving Systems and Testing Consistency

Write the system as $AX = B$. If $\lvert A \rvert \ne 0$, there is a unique solution $X = A^{-1}B$.

Worked example: Solve $2x + y = 5$ and $5x + 3y = 13$. Using the inverse above, $X = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\begin{pmatrix} 5 \\ 13 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$, so $x = 2$ and $y = 1$.

If $\lvert A \rvert = 0$, examine $(\operatorname{adj} A)B$. If it is not the zero matrix, the system has no solution. If it is the zero matrix, the system is either consistent with infinitely many solutions or inconsistent, so check by row reduction.

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Common Traps to Avoid

• Using the wrong sign pattern in cofactor expansion.
• Writing $(AB)^{-1} = A^{-1}B^{-1}$. The correct order is $B^{-1}A^{-1}$.
• Forgetting that $\lvert kA \rvert = k^n \lvert A \rvert$, not $k\lvert A \rvert$.
• Taking the area of a triangle as the determinant value without the factor $\tfrac{1}{2}$ and absolute value.

60-Second Revision Sheet

• $\lvert AB \rvert = \lvert A \rvert\lvert B \rvert$; $\lvert kA \rvert = k^n\lvert A \rvert$; $\lvert \operatorname{adj} A \rvert = \lvert A \rvert^{n-1}$
• $A^{-1} = \operatorname{adj}A / \lvert A \rvert$; $(AB)^{-1} = B^{-1}A^{-1}$; $(AB)^T = B^TA^T$
• Unique solution if $\lvert A \rvert \ne 0$: $X = A^{-1}B$
• Collinear points: the area determinant equals $0$

Your Study Plan

1. Day 1: matrix types, addition, multiplication and transpose.
2. Day 2: determinant evaluation using row operations and properties.
3. Day 3: adjoint, inverse and identity-based problems.
4. Day 4: linear systems, consistency tests and a timed mixed set.

Practice Matrices & Determinants Questions Free → (opens in a new tab)

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Continue Your Maths Journey

• Previous chapter: Complex Numbers & Quadratic Equations
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• All 14 JEE Main Maths chapters
• Complete JEE Main Syllabus 2027 guide

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