Square Matrix Properties

What a square matrix is

A square matrix has the same number of rows and columns, $n \times n$. Only square matrices have a determinant, an inverse, a main diagonal (the entries $a_{11}, a_{22}, \ldots, a_{nn}$) and a trace (the sum of that diagonal).

The square of a matrix

$A^2$ means $A \times A$ using matrix multiplication (each entry is a row of $A$ times a column of $A$), not squaring each entry. It's only defined for square matrices, since the number of columns of the first factor must match the rows of the second.

Example: find $A^2$ for $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

Squaring each entry instead would give $\begin{bmatrix} 1 & 4 \\ 9 & 16 \end{bmatrix}$, which is a different (and wrong) answer.

Scaling a determinant

Multiplying every entry of an $n \times n$ matrix by a constant $k$ scales the determinant by $k^n$, not just $k$:

If $A$ is a $3\times3$ matrix with $|A|=5$, then $|2A| = 2^3(5) = 40$.

Determinant properties of square matrices

Example: $A$ is $3 \times 3$ with $|A| = 4$. Then $|\text{adj }A| = 4^{2} = 16$, $|A^{-1}| = \dfrac{1}{4}$ and $|3A| = 3^3 \times 4 = 108$.

Worked example: an idempotent matrix

If $A$ is a square matrix with $A^2 = A$ (called idempotent), simplify $(I-A)^3 + A$.

Solution: First find $(I-A)^2$:

So squaring $(I-A)$ gives back $(I-A)$ itself: $(I-A)$ is idempotent too. That means:

So $(I-A)^3 + A = (I-A) + A = I$.

Building a matrix from a formula

For a $2\times2$ matrix $A = [a_{ij}]$ where $a_{ij} = \dfrac{i}{j}$, find $A$.

Solution: Compute each entry directly from its row/column index:

A singular matrix and its system of equations

For a matrix equation $AX=B$ with square matrix $A$:

• If $|A| \neq 0$, the system has exactly one solution, $X = A^{-1}B$.
• If $|A|=0$ (A is singular) and $(\text{adj }A)B \ne O$, the system has no solution at all: it's inconsistent.
• If $|A| = 0$ and $(\text{adj }A)B = O$, the test doesn't decide. The system may have infinitely many solutions or none, so it has to be checked directly, for example by elimination.

Common mistakes

• Squaring each entry. $A^2$ is $A$ multiplied by itself as matrices, as the example above shows.
• Writing $|kA| = k|A|$. Each of the $n$ rows picks up a factor of $k$, so it's $k^n|A|$.
• Assuming $AB = BA$. Matrix multiplication isn't commutative, so $(A + B)^2 = A^2 + AB + BA + B^2$, not $A^2 + 2AB + B^2$.
• Treating $(\text{adj }A)B = O$ as "infinitely many solutions". With $|A| = 0$ it can also mean no solution.

Where this comes up in exams

Determinants of square matrices and the adjoint and inverse are listed in the matrices units of the JEE Main 2026 and JEE Advanced 2026 syllabi and the 2025-26 CBSE Class XII syllabus. JEE Main and CBSE also name the consistency of a system of linear equations, and JEE Advanced its solution.

Checked against JEE (Main) 2026 syllabus (NTA), JEE (Advanced) 2026 syllabus and CBSE Mathematics curriculum, Classes XI-XII (2025-26). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

• Fit it into your week with the free timetable generator.
• Mark it off as you revise in the syllabus tracker.
• See how long you have left: JEE Main 2027 (Session 1) countdown and JEE Advanced 2027 countdown.