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# NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 Determinants - PDF Download

NCERT solutions for class 12 maths chapter 4 exercise 4.5 Determinants provides deep understanding of applications of determinants and matrices in solving the system of linear equations in two or three variables. The questions included in these solutions are excellent for providing a thorough understanding of using determinants to achieve linear equation solutions. Class 12 maths chapter 4 NCERT solutions are also important for developing the error-free learning strategy needed to perform well in board exams.

You will find a total of 16 questions in ex 4.5 class 12 maths chapter 4 that delivers an in-depth understanding of finding solutions of linear equations using matrices. Ex 4.5 class 12 maths ch 4 NCERT solutions are developed by the subject experts of eSaral to help students get clear understanding of concepts so that they can achieve good marks in exams.

NCERT solutions for ex 4.5 class 12 maths ch 4 is provided here in PDF format. You can download the free PDF NCERT solutions from the official website of eSaral and practice with the questions and concepts offline anytime anywhere.

## Topics Covered in Exercise 4.5 Class 12 Mathematics Questions

Ex 4.5 class 12 maths solutions covers some important topics like applications of determinants and matrices, solution of system of linear equations using inverse of a matrix. Check the detailed solutions of these topics which are explained by our experts of eSaral.

 1 Applications of Determinants and Matrices Solution of system of linear equations using inverse of a matrix
1. Applications of Determinants and Matrices

• Systems of linear equations in two or three variables can be solved with these applications of determinants and matrices. They can also be used to verify that a system of linear equations is consistent.

• Consistent system: A system of equations is said to be consistent if its solution (one or more) exists.

• Inconsistent system: A system of equations is said to be inconsistent if its solution does not exist.

• We can define the determinant as a number that establishes whether a system of linear equations has a unique solution.

• Solution of system of linear equations using inverse of a matrix

Let's use the inverse of the coefficient matrix to solve the system of linear equations expressed as matrix equations.

Consider the system of equations

\begin{aligned} & a_1 x+b_1 y+c_1 z=d_1 \\ & a_2 x+b_2 y+c_2 z=d_2 \\ & a_3 x+b_3 y+c_3 z=d_3\end{aligned}

Let $\mathrm{A}=\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right], \mathrm{X}=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}d_1 \\ d_2 \\ d_3\end{array}\right]$

Then, the system of equations can be written as, AX = B, i.e.,

$\left[\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}d_1 \\ d_2 \\ d_3\end{array}\right]$

Case I -  If A is a nonsingular matrix, then its inverse exists.

Now AX = B

A–1 (AX) = A–1 B (pre multiplying by A–1)

(A–1A) X = A–1 B (by associative property)

I X = A–1 B

X = A–1 B

This matrix equation provides a unique solution for the given system of equations as the inverse of a matrix is unique. This method of solving the system of equations is known as the Matrix Method.

Case II -  If A is a singular matrix, then |A| = 0

In this case, we calculate (adj A) B.

If (adj A) B ≠ O, (O being zero matrix), then the solution does not exist and the system of equations is called inconsistent.

If (adj A) B = O, then system may be either consistent or inconsistent as the system have either infinitely many solutions or no solution.

## Tips for Solving Exercise 4.5 Class 12 Chapter 4 Determinants

eSaral’s subject experts have included some effective tips to solve questions of ex 4.5 class 12 maths chapter 4.

1. Students should first solve simple questions then move on to complex questions of ex 4.5 with step by step solutions.

2. It is advised that in order to gain a thorough understanding, one should practice studying complex concepts repeatedly.

3. You must go through the fundamental concepts of ex 4.5 to be well versed with the questions.

## Importance of Solving Ex 4.5 Class 12 Maths Chapter 4 Determinants

Solving ex 4.5 class 12 maths chapter 4 will provide you a lot of benefits.

1. By solving questions of ex 4.5 class 12 maths chapter 4 NCERT solutions, students will be able to gain a thorough comprehension of all difficult topics.

2. Class 12 math chapter 4 exercise 4.5 NCERT solutions provide clear and concise answers to all the questions in developing solid problem-solving skills for board exams.

4. All the questions are solved by subject experts of eSaral in stepwise method so that you can comprehend the concepts used in each question.