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Class 12 Matrices: Complete One-Shot Revision Notes

This one-shot lecture covers fundamental matrix concepts for Class 12 mathematics, including definitions, order, and elements. It details matrix operations such as addition, scalar multiplication, and matrix multiplication conditions. Additionally, it explores special matrices like symmetric, skew-symmetric, and orthogonal matrices, as well as properties of transpose and trace.

Class 12 Science
Class 12 Commerce
Maths
English · 7 topics
Checked against the video on 25 Oct 2026

Made by AI from the YouTube video Class 12 Maths Chapter 3 Matrices One Shot | Detailed Explanation | MP Board 2026 | By Sanjeev Sir (opens YouTube in a new tab) by MP Board Wallah, then checked against it. All credit for the teaching goes to the creator. Tap a topic’s time to confirm a fact in the video.

Class 12 Matrices: Complete One-Shot Revision Notes

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In short

This one-shot lecture covers fundamental matrix concepts for Class 12 mathematics, including definitions, order, and elements. It details matrix operations such as addition, scalar multiplication, and matrix multiplication conditions. Additionally, it explores special matrices like symmetric, skew-symmetric, and orthogonal matrices, as well as properties of transpose and trace.

Definition and Order of a Matrix

A matrix is an ordered rectangular array of numbers, functions, or characters arranged in horizontal rows and vertical columns.
Individual numbers, functions, or characters inside a matrix are called its elements or entries, and matrices are denoted by capital letters.
Must knowThe order of a matrix indicates its dimensions and is written as m × n (read as m by n), where m is the number of rows and n is the number of columns.
The total number of elements in a matrix of order m × n is equal to the product of m and n.
Matrices are conventionally enclosed within square brackets [ ].

Matrix Element Notation and Possible Orders

Elements are designated by their position a_ij, where i denotes the specific row number and j denotes the specific column number.
Must knowIf a matrix has a prime number of elements p, it can only have exactly 2 possible orders: 1 × p and p × 1.
For a composite count like 24 elements, all factor pairs (m, n) such that m × n = 24 give the possible orders (8 total orders).
The total number of possible matrices of order m × n with a fixed number of choices per entry is given by (number of choices)^{m × n}.

Types of Matrices

A row matrix contains only a single row (m = 1) and any number of columns (n ∈ N).
A column matrix contains only a single column (n = 1) and any number of rows (m ∈ N).
Must knowA square matrix is a matrix where the number of rows equals the number of columns (m = n).
A zero matrix (or null matrix) has all its elements equal to zero and can be of any order.
A diagonal matrix is a square matrix whose non-diagonal elements are all zero.
A scalar matrix is a diagonal matrix whose principal diagonal elements are all equal.
An identity matrix is a scalar matrix whose principal diagonal elements are all equal to 1.

Matrix Type	Row Condition	Column Condition	Diagonal Elements
Row Matrix	m = 1	n ∈ N	N/A
Column Matrix	m ∈ N	n = 1	N/A
Square Matrix	m = n	m = n	Any
Diagonal Matrix	m = n	m = n	Non-diagonal entries are 0
Scalar Matrix	m = n	m = n	Diagonal entries are equal, others 0
Identity Matrix	m = n	m = n	Diagonal entries are 1, others 0

Equality of Matrices and Matrix Operations

Two matrices are equal if they have the same order and their corresponding elements are identical (a_ij = b_ij).
Matrix addition and subtraction are defined only when both matrices have the exact same order, performed by adding or subtracting corresponding entries.
Multiplying a matrix by a scalar k requires multiplying every single entry of the matrix by k.
Must knowMatrix multiplication of A_(m×n) and B_(p×q) is defined if and only if the number of columns in A equals the number of rows in B (n = p).
The resulting product matrix C = AB has dimensions m × q.
Matrix multiplication is generally not commutative (AB ≠ BA), but diagonal matrices of the same order commute under multiplication.

Transpose of a Matrix and Orthogonal Matrices

The transpose of a matrix A (denoted as A' or A^T) is obtained by interchanging its rows and columns.
The double transpose of a matrix returns the original matrix: (A')' = A.
The reversal law for transpose states that (AB)' = B'A'.
Must knowA square matrix A is defined as orthogonal if AA' = A'A = I.

Symmetric and Skew-Symmetric Matrices

A square matrix A is symmetric if its transpose equals the original matrix (A' = A).
Must knowA square matrix A is skew-symmetric if its transpose equals the negative of the matrix (A' = -A).
All principal diagonal elements of a skew-symmetric matrix are always zero (a_ii = 0).
Every square matrix A can be uniquely expressed as the sum of a symmetric matrix P and a skew-symmetric matrix Q: A = P + Q, where P = 1/2(A + A') and Q = 1/2(A - A').
If a matrix is both symmetric and skew-symmetric simultaneously, it must be a zero matrix.

Trace of a Matrix and Invertible Matrices

The trace of a square matrix A, denoted Tr(A), is the sum of its principal diagonal elements.
The trace of any skew-symmetric matrix is always 0 because all its diagonal elements are zero.
Two matrices A and B are inverses of each other if and only if AB = BA = I.

Key terms

Order of a matrix: The dimensions m × n representing m rows and n columns
Square Matrix: A matrix having an equal number of rows and columns (m = n)
Diagonal Matrix: A square matrix where all non-diagonal elements are 0
Scalar Matrix: A diagonal matrix in which all diagonal elements are equal
Identity Matrix: A diagonal matrix where all diagonal elements equal 1
Transpose of Matrix: A new matrix formed by interchanging rows and columns, denoted A' or A^T
Symmetric Matrix: A square matrix where A' = A
Skew-Symmetric Matrix: A square matrix where A' = -A
Orthogonal Matrix: A matrix satisfying the condition AA' = A'A = I
Trace of Matrix: The sum of all elements lying along the principal diagonal of a square matrix

Formulas

Number of elements = m × n: Calculates the total entries in a matrix of order m × n
Total matrices = (choices)^{m × n}: Computes possible matrices given order and entry options
P = 1/2(A + A'): Finds the symmetric component of a square matrix A
Q = 1/2(A - A'): Finds the skew-symmetric component of a square matrix A
Tr(A) = Σ a_ii: Sum of principal diagonal elements of square matrix A

Quick revision

Read these the night before the exam.

The principal diagonal elements of any skew-symmetric matrix are always zero.
Matrix multiplication AB is possible only when columns of A equal rows of B.
Transpose product reversal rule is (AB)' = B'A'.
A matrix is orthogonal when AA' = A'A = I.
A single element matrix [k] is simultaneously a row matrix, column matrix, and square matrix.
Any square matrix can be written as A = 1/2(A + A') + 1/2(A - A').
If a matrix is both symmetric and skew-symmetric, it is a zero matrix.
For any skew-symmetric matrix, Tr(A) = 0.
Two matrices A and B are inverses of each other if and only if AB = BA = I.

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