RD Sharma Class 12 Solutions Chapter 5 - Algebra of Matrices (Ex 5.1) Exercise 5.1

This page contains RD Sharma Solutions for Class 12 Maths Chapter 5 – Algebra of Matrices. The most important thing for students to do to have a score well in Mathematics is to solve the questions of every exercise. RD Sharma Solutions for Class 12 is prepared by a team of experts who work to the best of their abilities to help students succeed in their exams. Students who practice these solutions regularly undoubtedly improve their skills, which are essential for good results.

Algebra of Matrices is a branch of mathematics that deals with vector spaces of various dimensions. The invention of matrix algebra arose as a result of the presence of n-dimensional planes in our coordinate space.

A matrix (plural: matrices) is a rectangular array of numbers, expressions, or symbols. This layout is done in horizontal rows and vertical columns in the order of several rows x number of columns. Every point pair in a three-dimensional space represents a distinct equation with one or more solutions.

Linear Algebra is the fundamental concept or central idea of applied mathematics. For example, by reliving the rules and regulations, or Axioms, we get into a generalization of vector space, which leads to the Solution of Differential Equations via calculus.

Matrix Algebra

Matrix algebra entails matrices' operations such as addition, subtraction, multiplication, and so on.

Let us gain a better understanding of the matrix's operation-

Matrices Addition/Subtraction

Two matrices can be added or subtracted if (and only if) the number of rows and columns in both matrices is the same, or the matrices are in the same order.

To add/subtract, each element of the first matrix is added/subtracted from the elements of the second matrix.

Multiplication of Matrixes

Matrix, for example, can be multiplied twice.

(i) Scalar Multiplication 

(ii) Matrix multiplication with a different matrix

Scalar Multiplication is the process of multiplying a scalar quantity by a matrix. To create a new matrix, multiply each element in the matrix by the scalar quantity.

Matrix Algebra Rule

Matrix algebra follows some rules for addition and multiplication. Consider A, B, and C to be three distinct square matrices. A' is the inverse of A, and A-1 is the transpose of A. R is a real number, and I is the identity matrix.

Now, according to the laws of matrices:

A+B = B+A →Commutative Law of Addition

A+B+C = A +(B+C) = (A+B)+C →Associative law of addition

ABC = A(BC) = (AB)C →Associative law of multiplication

A(B+C) = AB + AC →Distributive law of matrix algebra

R(A+B) = RA + RB

Also, see here rules for transposition of matrices:

(A’)’ = A

(A+B)’ = A’+B’

(AB)’ = B’A’

(ABC) = C’B’A’

The inverse rules of matrices are as follows:

• AI = IA = A

• AA-1 = A-1A = I

• (A-1)-1 = A

• (AB)-1 = B-1A-1

• (ABC)-1 = C-1B-1A-1

• (A’)-1 = (A-1)’

More Examples of Matrix Addition

Matrix addition is nothing more than a series of additions. 

Add the top left numbers together and write the sum in the top-left position of a new matrix.

Add the numbers in the top right and write the total in the top right.

Add the numbers in the bottom left and write the total in the bottom left.

Add the numbers in the bottom right and write the total in the bottom right.

What is a Matrix's Determinant?

A matrix's determinant is simply a special number that is used to describe matrices for solving systems of linear equations, finding inverse matrices, and other calculus applications. It's impossible to define in plain English; it's usually defined in mathematical terms or in terms of what it can help you do. A matrix's determinant has the following properties:

• It is a true figure. Negative numbers are included.

• Determinants are only available for square matrices.

• Only matrices with non-zero determinants have an inverse matrix.

• The symbol for the determinant of a matrix A is |A|, which is also the symbol for absolute value, even though the two are unrelated.

You can find RD Sharma solutions for class 12 maths chapter 5 on Vedantu. The topic of matrices algebra and its various types are covered in this chapter. A matrix is a rectangular array of numbers that is divided into columns and rows. The left matrix, for example, has two rows and three columns, whereas the ideal matrix has three rows and two columns. Matrices are used in a variety of applications and serve as the foundation for linear algebra. Their programs include solving systems of linear equations, graph theory path-finding, and a variety of group theory programs (particularly representation theory). They're also very useful for representing linear transformations (especially rotations) and thus complicated amounts.