Matrices

Important Questions

In ICSE Class 10 Mathematics, Matrices embarks on a journey into the intriguing realm of matrices—a mathematical tool with wide-ranging applications in fields such as science, engineering, and computer science. Matrices, essentially organized arrays of numbers in rows and columns, form the core of this chapter's exploration.

This chapter introduces students to the fundamental concepts of matrices and unveils their real-world significance. Matrices play pivotal roles in solving systems of linear equations, analyzing data, and creating computer graphics, making them indispensable in various disciplines. Through this chapter, students acquire a solid foundation in matrix basics, matrix operations, determinants, and their practical utility in tackling everyday problems.

Tables of Content

In the chapter matrices class 10, "Matrices" is a captivating subject that unlocks a potent mathematical tool. Picture matrices as organized grids, filled with numbers that hold specific positions in rows and columns. These grids may seem straightforward, but they wield immense power, finding applications in computer graphics, engineering, and practical problem-solving. As you delve into this chapter and solve matrices class 10 ICSE important questions you'll embark on a journey into the intricate world of matrices. You'll master the art of working with them, performing operations, and comprehend their profound significance. Matrices serve as the backbone for solving equations, dissecting data, and creating stunning visual effects in industries like entertainment and technology.

In the chapter matrices class 10 ICSE, a matrix, in its simplest form, is a rectangular arrangement or table of numbers, symbols, or expressions, organized into rows and columns. Each number or element within a matrix has a specific location, identified by its row and column. Matrices are fundamental mathematical objects used for various purposes, including solving systems of linear equations, data organization, computer graphics, and transformations in geometry.

Here are some key points to understand about matrices in ICSE Class 10 Mathematics:

**Organization:** A matrix is structured into rows and columns, much like a spreadsheet. The number of rows and columns in a matrix is referred to as its dimensions.

**Elements:** Each entry or element within a matrix is represented by a number or symbol. The elements of a matrix are typically denoted by letters with subscripts to indicate their position, such as "a₁₁" for the element in the first row and first column.

**Types of Matrices:** Matrices come in various forms, including square matrices (with an equal number of rows and columns), row matrices (with one row and multiple columns), and column matrices (with one column and multiple rows).

and A

(b) 30

(c) 36

(d) 40

**Ans**. (c) 36

**Explanation:** A

⇒ A, A,=B

⇒ \(\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}\)=\(\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}\)

⇒ \(\begin{bmatrix} 4 & 36 \\ 0 & 1 \end{bmatrix}\) = \(\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}\)

⇒ x = 36

(b) \(\begin{bmatrix} 2 & -2\\ 2 & 0 \end{bmatrix}\)

(c) \(\begin{bmatrix} 1 & -2 \\ 6 & 0 \end{bmatrix}\)

(d) \(\begin{bmatrix} -1 & 7 \\ 9 & 0 \end{bmatrix}\)

**Ans**. (a)

**Explanation:**we have,

AB = \(\begin{bmatrix} 2 & -1 \\ 2 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\)

=\(\begin{bmatrix} 2 & -2 \\ 2 & 0 \end{bmatrix}\)

\(\begin{bmatrix} 2 & -2 \\ 2 & 0 \end{bmatrix}\)+\(\begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix}\)

∴ AB + C =\(\begin{bmatrix} -1 & 0 \\ 6 & 0 \end{bmatrix}\)

is the product AB possible ? Give a reason. If yes, find AB.

**Explanation:**A \(\begin{bmatrix} 3 & 5 \\ 4 & -2 \end{bmatrix}_{2×2}\) and B = \(\begin{bmatrix} 2 \\ 4 \end{bmatrix}_{2×1}\)

The product AB is possible as the number of columns in A is equal to the number of rows in

B. Now AB = \(\begin{bmatrix} 3 & 5 \\ 4 & -2\end{bmatrix}\begin{bmatrix} 2\\ 4\end{bmatrix}\)

=\( \begin{bmatrix} 3×2+5×4\\ 4 × 2 + (-2) × 4\end{bmatrix}\)

= \(\begin{bmatrix} 26 \\ 0\end{bmatrix}\)

Given A = \(\begin{bmatrix} x & 3 \\ y & 3 \end{bmatrix}\)

**Explanation:**Also A

\(\Rightarrow \begin{bmatrix} x & 3 \\ y & 3 \end{bmatrix}\begin{bmatrix} x & 3 \\ y & 3 \end{bmatrix}\) = 3\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

\(\Rightarrow \begin{bmatrix} x^2+3y & 3x+9 \\ xy+3y & 3y+9 \end{bmatrix}\)= \(\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\)

Comparing both sides, we get

3x + 9 = 0

⇒ x= -\(\frac{9}{3}\)=-3

and 3y + 9 = 3

⟹ 3y = 3 – 9 = – 6

⟹y=-\(\frac{6}{3}\)=-2

∴

**Explanation:**LET A= \(\begin{bmatrix} 2 & 1 \\ 0 & -2\end{bmatrix}\), B= \(\begin{bmatrix} 4 & 1 \\ -3 & -2\end{bmatrix}\) and C= \(\begin{bmatrix} -3 & 2 \\ -1 & 4\end{bmatrix}\)

A

=\(\begin{bmatrix} 4+0 & 2-2 \\ 0 & 0+4\end{bmatrix}\)=\( \begin{bmatrix} 4 & 0 \\ 0 & 4\end{bmatrix}\)

5B= \(\begin{bmatrix} 20 & 5 \\ -15 & -10\end{bmatrix}\)

AC = \(\begin{bmatrix} 2 & 1 \\ 0 & -2\end{bmatrix}\begin{bmatrix} -3 & 2 \\ -1 & 4\end{bmatrix}\)

=\(\begin{bmatrix} -6-1 & 4+4 \\ 0+2 & 0-8\end{bmatrix}\) = \(\begin{bmatrix} -7 & 8 \\ 2 & -8\end{bmatrix}\)

∴ A

=\(\begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}\)+\(\begin{bmatrix} -7 & 8 \\ 2 & -8 \end{bmatrix}\)-\(\begin{bmatrix} 20 & -5 \\ 2 & -8 \end{bmatrix}\)

=\(\begin{bmatrix} 4-7-20 & 0+8-5 \\ 0+2+15 & 4-8+10 \end{bmatrix}\)

=\(\begin{bmatrix} -23 & 3 \\ 17 & 6 \end{bmatrix}\)

The study of matrices class 10 ICSE is both essential and versatile, finding practical applications across diverse fields. Matrices, organized arrays of numbers in rows and columns, serve as a potent tool for equation solving, data manipulation, and geometric transformations. In this chapter, students have delved into the foundational concepts of matrices, grasping their structure, elements, operations, and real-world uses. They've acquired proficiency in matrix operations like addition, subtraction, and multiplication, comprehending the governing rules. If you're aiming to boost your proficiency by practicing more and gaining a thorough grasp of the chapter's content, oswal.io provides a comprehensive array of matrices class 10 ICSE questions to assist you in your pursuit of a deeper understanding of the subject matter.

**Ans: **A matrix is an ordered arrangement of numbers or symbols in rows and columns.

**Ans: ** The dimensions of a matrix are given as "m × n," where "m" represents the number of rows, and "n" represents the number of columns.

**Ans: **Matrix addition is done by adding corresponding elements of two matrices of the same dimensions.

**Ans:** The order of a matrix is determined by its dimensions, specified as "number of rows x number of columns" (m x n). It signifies the size and structure of the matrix, crucial for various matrix operations and calculations in linear algebra. For example, a "3 x 2" matrix has 3 horizontal rows and 2 vertical columns.

**Ans:** Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix, following specific rules.