Lakshya JEE 2026 Batch Mathematics Matrices Lecture 01 Study Notes

Lakshya JEE 2026 Batch: Mathematics - Matrices

Course Information

• Subject: Mathematics

• Topic: Matrices

• Lecture: 01

• Instructor: Ashish Agarwal Sir (IIT Kanpur)

• Batch: Lakshya JEE 2026

Topics Covered in Lecture 01

1. Introduction to Matrices

2. Special types of Matrices

3. Determinant of a Square Matrix

4. Algebra of Matrices

Homework Discussion and TAH (Take-Away Homework) Solutions

TAH 02: Arithmetic Progression and Determinants

• Question: If $p, q, r, s$ are in A.P. and $f(x) = \begin{vmatrix} p+\sin(x) & q+\sin(x) & p-r+\sin(x) \\ q+\sin(x) & r+\sin(x) & r-s+\sin(x) \\ r+\sin(x) & s+\sin(x) & s-q+\sin(x) \end{vmatrix}$ such that $\int_0^2 f(x)\,dx = -4$, find the common difference of the A.P.

• Solution Strategy:

  • Let the common difference be $d$.

  • Perform row transformations: $R_1 \rightarrow R_1 - R_2$ and $R_2 \rightarrow R_2 - R_3$.

  • Since $p, q, r, s$ are in A.P., then $p-q = -d$, $q-r = -d$, $r-s = -d$, etc.

  • The determinant value simplifies to $f(x) = -2d^2$.

  • Integrate: $\int_0^2 -2d^2\,dx = -2d^2 [x]_0^2 = -4d^2$.

  • Set equal to $-4$: $-4d^2 = -4 \Rightarrow d^2 = 1 \Rightarrow d = \pm 1$.

• Options: A) -1, B) 1/2, C) 1, D) 2

• Correct Answer: A and C ($d = 1$ or $d = -1$).

TAH 03: Function Differentiation

• Question: Let $F(x) = \begin{vmatrix} 1 & 1+\sin(x) & 1+\sin(x)+\cos(x) \\ 2 & 3+2\sin(x) & 4+3\sin(x)+2\cos(x) \\ 3 & 6+3\sin(x) & 10+6\sin(x)+3\cos(x) \end{vmatrix}$. Find $F'(\frac{\pi}{2})$.

• Solution Strategy:

  • Perform row operations: $R_2 \rightarrow R_2 - 2R_1$ and $R_3 \rightarrow R_3 - 3R_1$.

  • The resulting determinant simplifies to the constant value $F(x) = 1$.

  • The derivative of a constant is zero: $F'(x) = 0$.

  • Thus, $F'(\frac{\pi}{2}) = 0$.

• Correct Answer: B (0).

TAH 04: IIT-JEE 1985 Combination Property

• Question: Show that $\Delta = \begin{vmatrix} ^xC_r & ^xC_{r+1} & ^xC_{r+2} \\ ^yC_r & ^yC_{r+1} & ^yC_{r+2} \\ ^zC_r & ^zC_{r+1} & ^zC_{r+2} \end{vmatrix} = \begin{vmatrix} ^xC_r & ^{x+1}C_{r+1} & ^{x+2}C_{r+2} \\ ^yC_r & ^{y+1}C_{r+1} & ^{y+2}C_{r+2} \\ ^zC_r & ^{z+1}C_{r+1} & ^{z+2}C_{r+2} \end{vmatrix}$.

• Principle Used: Pascal's Identity: $^nC_r + ^nC_{r+1} = ^{n+1}C_{r+1}$.

• Proof Steps:

  • Use column operations to transform entries. Adding column 1 to column 2 gives $^nC_r + ^nC_{r+1} = ^{n+1}C_{r+1}$.

  • Subsequent addition of modified columns results in the shown matrix form.

Introduction to Matrices

Definition

• A matrix is an arrangement of elements (numbers, symbols, or functions) in the form of rows and columns.

• Row: Horizontal alignment of elements.

• Column: Vertical alignment of elements.

• Mathematical Application: Matrices are primarily concerned with numbers as elements.

Difference Between Matrix and Determinant

1. Nature: A matrix is an arrangement/structure; a determinant is a numerical value associated with a square matrix.

2. Value: A matrix has no value; a determinant has a specific numerical value.

3. Rows/Columns: In a matrix, the number of rows may or may not be equal to the number of columns. In a determinant, the number of rows must equal the number of columns.

Representation and Order

• Notation: Matrices are represented using square brackets $[ \dots ]$ or parentheses $( \dots )$. Some older notations used double bars $|| \dots ||$, but this is now obsolete.

• Order: If a matrix A has $m$ rows and $n$ columns, its order is written as $m \times n$ (read as "m by n").

• Number of Elements: The total number of elements in a matrix of order $m \times n$ is $mn$.

• Abbreviated Form: $A = [a_{ij}]_{m \times n}$, where $a_{ij}$ represents the element in the $i$-th row and $j$-th column ($1 \le i \le m$ and $1 \le j \le n$).

• Real Matrix: If all elements of a matrix are real numbers, it is called a real matrix.

Constructing Matrices: Examples

• Example 1: Construct a $3 \times 2$ matrix where $a_{ij} = \frac{|i - 2j|}{2}$.

  • $a_{11} = 1/2, a_{12} = 3/2$

  • $a_{21} = 0, a_{22} = 1$

  • $a_{31} = 1/2, a_{32} = 1/2$

  • Matrix $A = \begin{pmatrix} 1/2 & 3/2 \\ 0 & 1 \\ 1/2 & 1/2 \end{pmatrix}$.

• Example 2: Find all possible orders for a matrix with 24 elements.

  • Factors of 24: $1 \times 24, 24 \times 1, 2 \times 12, 12 \times 2, 3 \times 8, 8 \times 3, 4 \times 6, 6 \times 4$. (8 possible matrices).

Special Types of Matrices

Basic Types

• Row Matrix: A matrix having only one row. Order is $1 \times n$. Also called a Row Vector.

• Column Matrix: A matrix having only one column. Order is $m \times 1$. Also called a Column Vector.

• Zero or Null Matrix: An $m \times n$ matrix where all entries are zero. Denoted by $O_{m \times n}$.

• Horizontal Matrix: A matrix where the number of columns is greater than the number of rows (n > m).

• Vertical Matrix: A matrix where the number of rows is greater than the number of columns (m > n).

Square Matrix and Related Sub-types

• Square Matrix: A matrix where the number of rows equals the number of columns ($m = n$).

  • Diagonal Elements: Entries $a_{ij}$ where $i = j$ (e.g., $a_{11}, a_{22}, a_{33}$).

  • Conjugate Elements: Entries $a_{ij}$ and $a_{ji}$ for $i \ne j$.

  • Total Diagonal Entries: $n$ for a matrix of order $n$.

  • Elements above Diagonal: Entries where i < j.

  • Elements below Diagonal: Entries where i > j.

• Triangular Matrix:

  • Upper Triangular Matrix (UTM): A square matrix where all entries below the main diagonal are zero ($a_{ij} = 0$ for all i > j).

  • Lower Triangular Matrix (LTM): A square matrix where all entries above the main diagonal are zero ($a_{ij} = 0$ for all i < j).

• Diagonal Matrix: A square matrix where all non-diagonal elements are zero ($a_{ij} = 0$ for $i \ne j$). Abbreviated as $diag(d_1, d_2, \dots, d_n)$.

• Scalar Matrix: A diagonal matrix where all diagonal entries are equal ($d_1 = d_2 = \dots = a$).

• Identity or Unit Matrix ($I$): A scalar matrix where all diagonal entries are equal to 1. Denoted as $I_n$.

Statistical Properties of Matrices of Order $n$

• Minimum number of zeroes in a Triangular Matrix: $\frac{n^2 - n}{2}$.

• Minimum number of cyphers (zeroes) in a Diagonal/Scalar/Unit Matrix: $n^2 - n$.

• Maximum number of zeroes in a Triangular Matrix: $\frac{n^2 + n - 2}{2}$.

Numerical Problem: Distinct Entries

• Question: $A$ is a square matrix of order $n$. Let $l$ be the maximum number of distinct entries if $A$ is triangular, $m$ be the max distinct entries if $A$ is diagonal, and $p$ be minimum zeroes if $A$ is triangular. If $l + 5 = p + 2m$, find $n$.

• Formulas:

  • $l = \frac{n(n+1)}{2} + 1$ (Diagonal entries + one set of zeros + above/below entries).

  • $m = n + 1$ (Diagonal entries + zero).

  • $p = \frac{n(n-1)}{2}$.

• Calculation: Substituting into $l + 5 = p + 2m$ reveals $n = 3$.

Determinant of a Square Matrix

• If $A$ is a square matrix, its determinant is denoted by $|A|$ or $\det(A)$.

• Singular Matrix: A square matrix $A$ such that $|A| = 0$.

• Non-Singular Matrix: A square matrix $A$ such that $|A| \ne 0$.

• Important Note: A null matrix always has a determinant of 0, but a matrix with a determinant of 0 is not necessarily a null matrix.

Advanced Determinant Questions

• Singular Matrix Condition: If $A = \begin{pmatrix} 2-k &amp; -k \\ -1 &amp; 3-k \end{pmatrix}$ is singular, find $5k - k^2$.

  • $|A| = (2-k)(3-k) - (-1)(-k) = 0 \Rightarrow 6 - 5k + k^2 - k = 0 \Rightarrow k^2 - 6k + 6 = 0$.

  • From $k^2 - 6k + 6 = 0$, we find $k^2 - 5k = k - 6$. Calculation leads to specific result (Ans: 4).

• Summation of Determinants (WB JEE 2023): If $M_r = \begin{pmatrix} r &amp; r-1 \\ r-1 &amp; r \end{pmatrix}$, find $\sum_{r=1}^{2008} \det(M_r)$.

  • $\det(M_r) = r^2 - (r-1)^2 = r^2 - (r^2 - 2r + 1) = 2r - 1$.

  • $\sum_{r=1}^{2008} (2r-1) = 2(\frac{2008 \times 2009}{2}) - 2008 = 2008^2$.

Algebra of Matrices

Equality of Matrices

Two matrices $A$ and $B$ are equal if and only if:

1. They have the same order ($O(A) = O(B)$).

2. Every corresponding element is identical ($a_{ij} = b_{ij}$ for all $i, j$).

Addition and Subtraction

• Defined only for matrices of the same order.

• Addition: $A + B = [a_{ij} + b_{ij}]$.

• Subtraction: $A - B = [a_{ij} - b_{ij}]$.

• Properties:

  1. Commutative: $A + B = B + A$.

  2. Associative: $A + (B + C) = (A + B) + C$.

  3. Additive Inverse: The additive inverse of $A$ is $-A = [-a_{ij}]$.

Questions & Discussion

Question Answered by Sakil (West Bengal)

• Context: Solution to the TAH 01 determinant problem involving $a^2+b^2+c^2 = 1$.

• Result: Showed that the determinant simplifies to $\cos^2(\theta)$.

Question Answered by Dhairya Goel (Faridabad)

• Context: Solution for the AP determinant (TAH 02) and the differentiation problem (TAH 03).

• Process: Demonstrated row/column reduction steps and the application of $\int f(x)\,dx$.

Question Answered by Ajay (Giridih, Jharkhand)

• Context: Verification of the combinations identity proof (TAH 04) using column operations.

Question Answered by Yash Singh (Jharkhand)

• Context: Provided calculation details for the row reduction and integration of the AP problem.