JEE Main Study Notes for Matrices and Determinants: Matrices and Determinants are about arranging or creating the objects, numbers, or alphabets in a rectangular array. Matrices and Determinants are very important topics in JEE Main 2022. This topic includes some topics such as algebra and type of matrices, matrices of order two and three, properties of determinants, the area of triangles using determinants, etc.

• JEE Main Matrices and Determinants has 6%-10% weightage in JEE Main Mathematics Syllabus.
• JEE Main Mathematics Question Paper is expected to consist of a total of around 7-10 marks from the topic Matrices and Determinants.

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What are Matrices?

A matrix is an arrangement of elements (numbers, mathematical expressions) in a rectangular arrangement along rows and columns. ‘m’ is the number of rows and ‘n’ is the number of columns.

Types of Matrices

• Row Matrix

A matrix in which all elements are arranged in a single row is a row matrix.

• Column Matrix

A matrix in which all elements are arranged in a single column is a column matrix.

• Square Matrix

A matrix in which all elements are arranged in an equal number of rows and columns, i.e. m=n, is called a square matrix.

The principal diagonal or leading diagonal of the matrix: The elements in a square matrix for which the number of the row is equal to the number of the column.

Trace of the matrix: The sum of the principal diagonal elements of a square matrix.

• Idempotent Matrix

A square matrix is an idempotent matrix if the product of the matrix itself results in the same matrix.

• Diagonal Matrix

A matrix is a diagonal matrix if all elements except those in the leading diagonal are zero.

• Scalar Matrix

A diagonal matrix in which all elements of the leading diagonal are equal will be a scalar matrix.

• Unit Matrix

A diagonal matrix in which all elements of the leading diagonal are equal is called a unit matrix

• Null Matrix

A matrix in which all elements are zero is called a null matrix.

• Symmetric Matrix

A matrix in which the element of the ith row and jth column is equal to the element of the jth row and ith column is called a symmetric matrix.

• Skew-symmetric Matrix

A matrix in which the element of the ith row and jth column is equal to the negative of the element of the jth row and ith column, such that all elements of the principal diagonal are zero.

• Triangular Matrix

1. a) Upper triangular Matrix: A matrix in which all elements below the principal diagonal are zero.
2. b) Lower triangular Matrix: A matrix in which all elements above the principal diagonal are zero.

Check: JEE Main Study Notes for Mathematical Reasoning

Algebra of Matrices

(a) Elementary Transformations of a matrix

(i) Interchange of columns/rows

(ii) Multiplication of a column or row by a non-zero number

(iii) The addition/subtraction of a constant multiple of the elements of any row/column to the corresponding elements of any other row/column

(b) Equivalent Matrices: If one of the matrices can be obtained by elementary transformations on the other then the two matrices are said to be equivalent

(c) Equal Matrices: Two matrices are said to be equal if and only if

(i) The order of the matrices is the same

(ii) The corresponding elements of matrices are equal

(d) Addition/Subtraction of Matrices: Two matrices can be added or subtracted if and only if the order of the matrices is the same. The resultant matrix will be the addition/subtraction of the corresponding elements.

(e) Multiplication of a matrix and a scalar: When a scalar is multiplied by a matrix, the resultant matrix is the scalar multiplied by each of the corresponding elements of the matrix.

(f) Multiplication of two matrices: Two matrices can be multiplied if and only if the number of rows in the first matrix is equal to the number of columns in the second matrix.

Also, note that it is not necessary that if two matrices A and B are multiplied then AB = BA.

(g) Transpose of a matrix: The transpose of a given matrix is the matrix obtained by interchanging the elements of columns and rows.

For a Symmetric matrix: A = AT

For a Skew-symmetric matrix: A= - AT

Note:

(i) The transpose of a product of the two matrices is always taken in reverse order i.e. (AB)T=BT AT

(ii) The square matrix can be expressed as the sum of a skew-symmetric matrix and

symmetric matrix

A = ½ (A + AT) + ½ (A - AT)

Here, { ½ (A + AT) } is a symmetric matrix while { ½ (A - AT) } is a skew-symmetric matrix.

Question:

Solution: 

Must Read:

• JEE Main Study Notes for Algebra
• JEE Main Study Notes for Sequence & Series

Determinant of a Matrix

A determinant for a given matrix exists only if it is a square matrix. It results in a single mathematical expression or number. It is denoted as |A|. It is evaluated as the sum of the products of elements of any row/column with its corresponding cofactor. For example, for matrix X of order 3,

Question: 

Solution: 

Properties of Determinants

1. If rows are changed into columns and columns into rows, then the values of the determinant remain unaltered.
2. The resulting determinant will be the negative of the original determinant if any two rows or columns of a determinant are interchanged,
3. If two rows or two columns in a determinant have corresponding elements that are equal, the value of the determinant is zero.
4. The determinant can be written as the sum of two or more determinants if each element in a row or column of a determinant is written as the sum of two or more terms.
5. If to each element of a line of a determinant, some multiples of corresponding elements of one or more parallel lines are added, then the determinant remains unchanged.
6. If each element in any row or column of determinant is zero, then the value of determinant is equal to zero.
7. If a determinant X vanishes for y = a, then (y - a) is a factor of X. In other words, if two rows or two columns become identical for y = a, then (y-a) is a factor of X.
8. The minor of an element of a determinant will be a determinant of lesser order which is formed by excluding the row and column of the element.

Question: 

Solution: Multiply row first, second, and third by a, b, c respectively, in the LHS then,

Take abc common from the third column

Interchange column first and third

Again interchange column second and third

= RHS

Question: Without expansion show that

Solution: In the LHS taking 2 commons from the fourth row, so

Since rows 2nd and 3rd are identical, so

= 2(0) = 0. Therefore, LHS=RHS

Must Read:

• JEE Main Study Notes for Probability
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Cofactors, Adjoint, and Inverse of a Matrix

Note that if the determinant formed by the cofactors of the corresponding elements of a matrix is equal to the square of the determinant of the matrix, then,

(a) Singular matrix

A matrix whose determinant is zero is a singular matrix

(b) Cofactor

Suppose there is an n-rowed square matrix A = [aij]. Then the cofactor Aij of aij is equal to (-1)i+j times the minor Mij of aij, i.e.,

(c) Adjoint of a matrix

The transpose of the matrix formed by taking the cofactors of each element to form a matrix is called the adjoint of the matrix. It is denoted by adj A for a matrix A.

Properties of the adjoint of matrix

If X is a square matrix of order n

(i) |adj X| = |X|n-1

(ii) adj (adj X) = |X|n-2 X,

(iii) |adj (adj X) | = |X|(n-2)(n-2)

(iv) adj (XY) = (adj Y) (adj X)

(v) adj (kX) = kn-1 (adj X), k is any scalar

(vi) adj XT = (adj X)T

Question: 

Solution: 

Inverse of a Matrix

The inverse of a non-singular matrix is a matrix that when multiplied by the original matrix results in an identity matrix.

Using adjoint, the inverse of a matrix can be calculated as 

Area of Triangles using Determinants

If the coordinates of the vertices of a triangle are (x1, y1), (x2, y2), and (x3, y3) then the area of the triangle is written as

Multiplication of Two Determinants

Let  and 

Then the multiplication of these two matrices is 

Check: JEE Main Study notes for Integral Calculus

Solution of Linear equations

1. Cramer’s Rule

Consider the system of equations

The determinant P is the determinant of the coefficient matrix. The determinant |Px| is the determinant obtained by replacing the column of coefficients of x with the column of constant terms. Similarly, we have |Py| and |Pz|

Then, the solution of the system of equations is defined as

x=|Px| / |P|

y=|Py| / |P|

z=|Pz| / |P|

2. Matrix Inversion method

The inversion of a matrix is given by

X=A-1B

Question: Use Cramer’s Rule to solve the system

Solution: 

Also Check: JEE Main Scalers and Vectors Study Notes

Questions for Practice

Tips to study Matrices & Determinants

Generally, the matrices and determinants questions are easy to solve and the chances of mistakes are less. However, candidates should take utmost importance while solving and avoid making silly errors while calculating. It is one of the scoring topics in JEE Main 2020 if the following tips are kept in mind:

1. The problems should be solved by the end so that you get enough practice of calculations too.
2. Candidates can make a separate sheet to revise all the properties. Many questions can directly be properties based.
3. The basic concepts and definitions should be memorized. Small questions can be made out of these.
4. Be mindful of what exactly is asked in the questions. The questions can be tricky sometimes.
5. Use a single standard book for solving. Avoid using a lot of material.

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