Introduction To Matrices

MathsMatricesFor JEE aspirants

A matrix is a rectangular array of numbers arranged in rows and columns, written inside square brackets $[]$ . In JEE Mathematics, matrices provide a compact language for handling systems of linear equations, geometric transformations, and computer graphics. This concept covers the definition of a matrix, its order, equal matrices, all major classifications (row, column, square, diagonal, scalar, identity, triangular, null), the trace of a square matrix, and the three fundamental operations that produce new matrices from old: transpose, conjugate, and transpose-conjugate.

Key Formulas - Quick Reference

Matrix of order $m \times n$ : $A = [a_{ij}]_{m \times n}$ , where $i$ is the row index and $j$ the column index.
Equal matrices: $A = B ⟺$ same order and $a_{ij} = b_{ij}$ for all $i, j$ .
Square matrix: $A = [a_{ij}]_{n \times n}$ (rows $=$ columns).
Trace of a square matrix: $tr (A) = i = 1 \sum n a_{ii} = a_{11} + a_{22} + \dots + a_{nn}$ .
Transpose: $A^{T} = [b_{ij}]_{n \times m}$ where $b_{ij} = a_{j i}$ . Properties: $(A^{T})^{T} = A$ , $(A \pm B)^{T} = A^{T} \pm B^{T}$ , $(α A)^{T} = α A^{T}$ , $(A B)^{T} = B^{T} A^{T}$ .
Conjugate $\overset{ˉ}{A}$ : replace each entry with its complex conjugate. $\overline{(\overset{ˉ}{A})} = A$ , $\overline{A + B} = \overset{ˉ}{A} + \overset{ˉ}{B}$ , $\overline{α A} = \overset{α}{ˉ} \overset{ˉ}{A}$ , $\overline{A B} = \overset{ˉ}{A} \overset{ˉ}{B}$ .
Transpose-conjugate: $A^{θ} = (\overset{ˉ}{A})^{T} = \overline{(A^{T})}$ . Properties: $(A^{θ})^{θ} = A$ , $(A + B)^{θ} = A^{θ} + B^{θ}$ , $(k A)^{θ} = \overset{ˉ}{k} A^{θ}$ , $(A B)^{θ} = B^{θ} A^{θ}$ .

1. What is a Matrix?

A rectangular array of symbols (real or complex numbers) arranged in rows and columns is called a matrix. A system of $m \times n$ symbols arranged in a rectangular formation along $m$ rows and $n$ columns, bounded by brackets $[]$ , is called an m by n matrix (written as $m \times n$ ).

The general form is:

$A = a_{11} a_{21} ⋮ a_{m 1} a_{12} a_{22} ⋮ a_{m 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{mn}_{m \times n}$

Here $a_{ij}$ is the element in the $i^{th}$ row and $j^{th}$ column.

Equal Matrices

Two matrices are said to be equal if they have the same order and each element of one is equal to the corresponding element of the other.

So $A = B$ requires (i) $A$ and $B$ have the same number of rows and same number of columns, and (ii) $a_{ij} = b_{ij}$ for every $(i, j)$ .

2. Classification of Matrices

Row Matrix

A matrix having a single row is called a row matrix.

Example: $[1357]$ is a $1 \times 4$ row matrix.

Column Matrix

A matrix having a single column is called a column matrix.

Example: $235$ is a $3 \times 1$ column matrix.

Square Matrix

An $m \times n$ matrix $A$ is called a square matrix if $m = n$ (number of rows equals number of columns).

Example: $A = 123234345$ is a square matrix of order $3 \times 3$ .

Note: The diagonal from the top-left corner to the bottom-right corner is called the leading diagonal or principal diagonal. In the example above, the leading diagonal contains the elements

1, 3, 5

Trace of a Matrix

The trace of a square matrix

A

is the sum of its principal-diagonal elements:

tr (A) = i = 1 \sum n a_{ii} = a_{11} + a_{22} + \dots + a_{nn}

Solved Example 1

Problem: Find the trace of

A = 123 3 - 1 1 464

Solution:

The principal-diagonal entries are $a_{11} = 1$ , $a_{22} = - 1$ , $a_{33} = 4$ .

$tr (A) = 1 + (- 1) + 4 = 4$

Diagonal Matrix

A square matrix all of whose elements outside the principal diagonal are zero is called a diagonal matrix. For $A = [a_{ij}]_{n \times n}$ to be diagonal: $a_{ij} = 0$ whenever $i \neq = j$ .

Example: $A = 300050 00 - 1$ is a diagonal matrix of order $3 \times 3$ .

Scalar Matrix

A diagonal matrix whose diagonal elements are all equal is called a scalar matrix. For $A = [a_{ij}]_{n \times n}$ to be scalar: $a_{ij} = {0 m i \neq = j i = j$ , where $m \neq = 0$ .

Example: $A = 500050005$ is a scalar matrix.

Unit Matrix (Identity Matrix)

A diagonal matrix of order $n$ with unity on all diagonal positions is called a unit matrix or identity matrix, denoted $I_{n}$ .

Formally: $a_{ij} = {10 i = j i \neq = j$ .

Example: $I_{3} = 100010001_{3 \times 3}$ .

Triangular Matrix

A square matrix in which all elements below the diagonal are zero is called an upper triangular matrix. A square matrix in which all elements above the diagonal are zero is called a lower triangular matrix.

Formally: $A = [a_{ij}]_{n \times n}$ is upper triangular if $a_{ij} = 0$ for $i > j$ , and lower triangular if $a_{ij} = 0$ for $i < j$ .

Notes:

A diagonal matrix is both upper and lower triangular.
A triangular matrix is called strictly triangular if all diagonal entries are also zero ( $a_{ii} = 0$ for $1 \leq i \leq n$ ).

Examples: $a 00 h b 0 g f c$ (upper triangular), $121 03 - 5 004$ (lower triangular).

Null Matrix (Zero Matrix)

If all elements of a matrix (square or rectangular) are zero, it is called a null or zero matrix. Formally: $a_{ij} = 0$ for all $i, j$ .

Example: $000000000$ is a $3 \times 3$ zero matrix.

Quick Classification Table

Type	Defining condition	Notation / example
Row	Single row (order $1 \times n$ )	$[135]$
Column	Single column (order $m \times 1$ )	$[2; 3; 5]^{T}$
Square	$m = n$	$3 \times 3$
Diagonal	Square + off-diagonal $= 0$	$diag (3, 5, - 1)$
Scalar	Diagonal + all diagonal entries equal	$diag (5, 5, 5)$
Identity	Scalar with diagonal $= 1$	$I_{n}$
Upper triangular	$a_{ij} = 0$ for $i > j$	Elements below diagonal zero
Lower triangular	$a_{ij} = 0$ for $i < j$	Elements above diagonal zero
Null / Zero	All $a_{ij} = 0$	$O$ or $0$

3. Transpose of a Matrix

The matrix obtained from

A

by interchanging its rows and columns is called the transpose of

A

, denoted

A^{T}

(or

A^{'}

). If

A = [a_{ij}]_{m \times n}

and

A^{T} = [b_{ij}]_{n \times m}

, then

b_{ij} = a_{j i}

for all

i, j

Example: If $A = 147258_{3 \times 2}$ , then $A^{T} = [124578]_{2 \times 3}$ .

Properties of Transpose

$(A^{T})^{T} = A$ - transpose of transpose is the original matrix.
$(A \pm B)^{T} = A^{T} \pm B^{T}$ (when $A$ and $B$ are of the same order).
$(α A)^{T} = α A^{T}$ (where $α$ is a scalar).
$(A B)^{T} = B^{T} A^{T}$ - note the order reversal (when $A$ and $B$ are conformable for multiplication).

4. Conjugate of a Matrix

The matrix obtained from a complex-valued matrix $A$ by replacing each entry with its complex conjugate is called the conjugate of $A$ , denoted $\overset{ˉ}{A}$ .

Example: If $A = 1 + 2 i 4 - 5 i 8 2 - 3 i 5 + 6 i 7 + 8 i 3 + 4 i 6 - 7 i 7$ , then $\overset{ˉ}{A} = 1 - 2 i 4 + 5 i 8 2 + 3 i 5 - 6 i 7 - 8 i 3 - 4 i 6 + 7 i 7$ .

Properties of Conjugate

$\overline{(\overset{ˉ}{A})} = A$ - conjugate of conjugate is the original.
$\overline{A + B} = \overset{ˉ}{A} + \overset{ˉ}{B}$ .
$\overline{α A} = \overset{α}{ˉ} \overset{ˉ}{A}$ for any complex scalar $α$ .
$\overline{A B} = \overset{ˉ}{A} \overset{ˉ}{B}$ (when $A$ and $B$ are conformable for multiplication).

5. Transpose Conjugate of a Matrix

The transpose of the conjugate of

A

is called the transpose-conjugate (or Hermitian conjugate) of

A

, denoted

A^{θ}

(also written

A^{*}

A^{H}

Equivalently, the conjugate of the transpose of $A$ is the same as the transpose of the conjugate: $\overline{A^{T}} = (\overset{ˉ}{A})^{T} = A^{θ}$ .

If $A = [a_{ij}]_{m \times n}$ , then $A^{θ} = [b_{j i}]_{n \times m}$ where $b_{j i} = \overline{a_{ij}}$ . In words: the $(j, i)$ -entry of $A^{θ}$ is the complex conjugate of the $(i, j)$ -entry of $A$ .

Example: If $A = 1 + 2 i 4 - 5 i 8 2 - 3 i 5 + 6 i 7 + 8 i 3 + 4 i 6 - 7 i 7$ , then $A^{θ} = 1 - 2 i 2 + 3 i 3 - 4 i 4 + 5 i 5 - 6 i 6 + 7 i 8 7 - 8 i 7$ .

Properties of Transpose-Conjugate

$(A^{θ})^{θ} = A$ .
$(A + B)^{θ} = A^{θ} + B^{θ}$ .
$(k A)^{θ} = \overset{ˉ}{k} A^{θ}$ (note the conjugation of the scalar).
$(A B)^{θ} = B^{θ} A^{θ}$ - order reversal, like transpose.

Solved Example 2

Problem: If

A = [2 + i 1 - i 3 - 2 i]

, find

A^{T}

\overset{ˉ}{A}

, and

A^{θ}

Solution:

Transpose (swap rows and columns):

$A^{T} = [2 + i 3 1 - i - 2 i]$

Conjugate (replace each entry with its conjugate):

$\overset{ˉ}{A} = [2 - i 1 + i 3 2 i]$

Transpose-conjugate (transpose of conjugate, or equivalently, conjugate of transpose):

$A^{θ} = [2 - i 3 1 + i 2 i]$

Common Mistakes to Avoid

Watch out

Confusing scalar matrix with identity matrix. A scalar matrix has all equal diagonal entries $m$ (any nonzero value). The identity is the special scalar matrix with $m = 1$ . Every identity is scalar, but not every scalar is the identity.
Forgetting the order reversal in $(A B)^{T}$ . $(A B)^{T} = B^{T} A^{T}$ , not $A^{T} B^{T}$ . Same reversal applies to inverse and transpose-conjugate.
Adding matrices of different orders. Matrix addition (and equality) requires the same order. A $2 \times 3$ and a $3 \times 2$ cannot be added, even though both have six entries.
Confusing transpose with inverse. $A^{T}$ swaps rows and columns; $A^{- 1}$ is the multiplicative inverse. These are equal only for orthogonal matrices, and in general are entirely different operations.
Not conjugating the scalar in $(k A)^{θ}$ . The rule is $(k A)^{θ} = \overset{ˉ}{k} A^{θ}$ , with $\overset{ˉ}{k}$ (the complex conjugate of $k$ ). For real $k$ , $\overset{ˉ}{k} = k$ , but for complex $k$ , forgetting this is a common error.
Computing trace of a non-square matrix. Trace is defined only for square matrices - a $2 \times 3$ or $3 \times 4$ matrix has no principal diagonal to sum.

Frequently Asked Questions

What is the order of a matrix?

The order of a matrix is written as $m \times n$ , where $m$ is the number of rows and $n$ is the number of columns. A matrix with 3 rows and 4 columns has order $3 \times 4$ , regardless of what values the entries take.

Can a matrix be simultaneously a row matrix and a column matrix?

Yes, but only if it is a $1 \times 1$ matrix (a single element). Any other row matrix has order $1 \times n$ with $n > 1$ (multiple columns, one row) and cannot also be a column matrix.

What is the trace of an identity matrix?

The trace of the identity matrix $I_{n}$ is $n$ , since every diagonal entry equals $1$ and there are $n$ of them. So $tr (I_{2}) = 2$ , $tr (I_{3}) = 3$ , and so on.

Is the transpose of a symmetric matrix equal to itself?

Yes, by definition. A matrix $A$ is symmetric if $A^{T} = A$ , which means $a_{ij} = a_{j i}$ for all $i, j$ . Symmetric matrices are covered in detail in the Algebra Of Matrices concept.

What is the difference between transpose and conjugate-transpose?

The transpose $A^{T}$ just swaps rows and columns without changing entry values. The transpose-conjugate $A^{θ}$ swaps rows and columns and also replaces each entry with its complex conjugate. For real matrices, $A^{T} = A^{θ}$ ; for complex matrices, they differ wherever any entry has a nonzero imaginary part.

Are diagonal matrices always square?

Yes. The definition of a diagonal matrix requires it to be square ( $m = n$ ), because the concept of a principal diagonal is defined only for square matrices. A rectangular matrix with zeros off the leading diagonal is sometimes called a rectangular diagonal matrix, but this is a less common term.

How is the trace related to the determinant?

For a $2 \times 2$ matrix $A$ , if $λ_{1}, λ_{2}$ are its eigenvalues, then $tr (A) = λ_{1} + λ_{2}$ and $det (A) = λ_{1} λ_{2}$ . More generally for an $n \times n$ matrix, the trace equals the sum of eigenvalues and the determinant equals their product. This ties matrix invariants to their spectral properties, an idea explored more deeply beyond the JEE level.

Why is the identity matrix denoted by $I$ ?

The letter $I$ stands for identity, because $I \cdot A = A \cdot I = A$ for any conformable matrix $A$ , mirroring how the number $1$ is the multiplicative identity for real numbers. This makes $I$ the multiplicative identity in the algebra of square matrices.

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