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What is the weightage of Matrices in JEE Mains Exam?

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Matrices is generally combined with determinants, Matrices and Determinants carry significant weightage in the competitive exams. Matrices carries a weightage of around 6-8% of the total marks in JEE Mains. Students can expect 1-2 questions in JEE Mains from this chapter. Students must have a clear conceptual understanding of the syllabus to perform better. Students can take help of NCERT soluition of Matrcies chapter to build strong foundation of this chapter.

There are several important Topics in Matrices for JEE Mains such as Types of Matrices, Operations on Matrices, Adjoint and Inverse of a Matrix, solving system of Linear Eq

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If $A [\begin{matrix} 1 & 0 & 1 \end{matrix}] = 2 [\begin{matrix} 1 & 0 & 1 \end{matrix}]$ , $A [\begin{matrix} - 1 & 0 & 1 \end{matrix}] = 4 [\begin{matrix} - 1 & 0 & 1 \end{matrix}]$ and $A [\begin{matrix} 0 & 1 & 0 \end{matrix}] = 2 [\begin{matrix} 1 & 0 & 0 \end{matrix}]$ where, A is a 3 × 3 matrix and (A –3I) $[\begin{matrix} x & y & z \end{matrix}] = [\begin{matrix} - 1 & 2 & 3 \end{matrix}]$ then the value of (x, y, z) is

alok kumar singh

Let $A = [\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix}]$

Given $A = [\begin{matrix} 1 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$ ...(1)

$[\begin{matrix} x_{1} + z_{1} & x_{2} + z_{2} & x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$

∴ x₁ + z₁ = 2 … (2)

x₂ + z₂ = 0 … (3)

x₃ + z₃ = 0 … (4)

Given $A = [\begin{matrix} - 1 & 0 & 1 \end{matrix}] = [\begin{matrix} - 4 & 0 & 4 \end{matrix}]$

$[\begin{matrix} - x_{1} + z_{1} & - x_{2} + z_{2} & - x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 4 & 0 & 4 \end{matrix}]$

⇒ – x₁ + z₁ = −4 … (5)

–x₂ + z₂ = 0 &nbs

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Let $A = [\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix}]$

Given $A = [\begin{matrix} 1 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$ ...(1)

$[\begin{matrix} x_{1} + z_{1} & x_{2} + z_{2} & x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$

∴ x₁ + z₁ = 2 … (2)

x₂ + z₂ = 0 … (3)

x₃ + z₃ = 0 … (4)

Given $A = [\begin{matrix} - 1 & 0 & 1 \end{matrix}] = [\begin{matrix} - 4 & 0 & 4 \end{matrix}]$

$[\begin{matrix} - x_{1} + z_{1} & - x_{2} + z_{2} & - x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 4 & 0 & 4 \end{matrix}]$

⇒ – x₁ + z₁ = −4 … (5)

–x₂ + z₂ = 0 … (6)

–x₃ + z₃ = 4 … (7)

Given $A = [\begin{matrix} 0 & 1 & 0 \end{matrix}] = [\begin{matrix} 0 & 2 & 0 \end{matrix}]$

$[\begin{matrix} y_{1} & y_{2} & y_{3} \end{matrix}] = [\begin{matrix} 0 & 2 & 0 \end{matrix}]$

∴ y₁ = 0, y₂ = 2, y₃ = 0

∴ from (2), (3), (4), (5), (6) and (7)

x₁ = 3, x₂ = 0, x₃ = –1

y₁ = 0, y₂ = 2, y₃ = 0

z₁ = –1, z₂ = 0, z₃ = 3

$A = [\begin{matrix} 3 & 0 & - 1 \\ 0 & 2 & 0 \\ - 1 & 0 & 3 \end{matrix}]$

Now (A – 3I) $[\begin{matrix} x & y & z \end{matrix}]$ = $[\begin{matrix} - 1 & 2 & 3 \end{matrix}]$

$[\begin{matrix} 0 & 0 & - 1 \\ 0 & - 1 & 0 \\ - 1 & 0 & 0 \end{matrix}] [\begin{matrix} x & y & z \end{matrix}] = [\begin{matrix} - 1 & 2 & 3 \end{matrix}]$

$[\begin{matrix} - z & - y & - x \end{matrix}]$ = $[\begin{matrix} - 1 & 2 & 3 \end{matrix}]$

[z = 1], [y = –2], [x = –3]

less

If the system of equations

kx + y + 2z = 1

3x – y – 2z = 2

-2x – 2y – 4z = 3

has infinitely many solutions, then k is equal to……………..

Vishal Baghel

kx + y + 2z = 1. (i)

3x – y – 2z = 2 . (ii)

2x – 2y – 4z = 3. (iii)

(ii) × 5 (i) $\equiv$ (iii) × 3 (15 – k) = 6

K = 21

Let A = $[\begin{matrix} x & y & z \\ y & z & x \\ z & x & y \end{matrix}]$ , where x, y and z are real numbers such that x + y + z > 0 and xyz = 2 If A² = I₃, then the value of x³ + y³ + z³ is…………………

Vishal Baghel

$A = [\begin{matrix} x & y & z \\ y & z & x \\ z & x & y \end{matrix}], | A | = 3 x y z - (x^{3} + y^{3} + z^{3}) = - (x + y + z) [{(x + y + z)}^{2} - 3 (x y + y z + z x)]$

A² = l

A. A’ = l (as A = A’)

$∴ x^{2} + y^{2} + z^{2} = 1 a n d x y + y z + z x = 0$

$x^{3} + y^{3} + z^{3} = 3 \times 2 + 1 \times (1 - 0) = 7$

Let P = $[\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}] w h e r e α \in R .$ Suppose Q = [q_ij] is a matrix satisfying PQ = kI₃ for some non-zero k $\in$ R. If $q_{23} = \frac{- k}{8} a n d | Q | = \frac{k^{2}}{2}, t h e n α^{2} + k^{2}$ is equal to_________.

Vishal Baghel

$P = [\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}] a n d Q = [q_{i j}] \Rightarrow P Q = k l_{3}$

$q_{23} = - \frac{k}{8} a n d | Q | = \frac{k^{2}}{2}$

$P Q = k l_{3} \Rightarrow P^{- 1} = \frac{Q}{k} = {(\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix})}^{- 1}$

$| p | | Q | = (k l_{3}) \Rightarrow 8 . \frac{k^{2}}{2} = k^{3}$

$k \neq 0 \Rightarrow k = 4$

$∴ α^{2} + k^{2} = 1 + 16 = 17 .$

For the system of linear equations:

x – 2y = 1, x – y + kz = -2, ky + 4z = 6, k $\in R,$

Consider the following statements:

(A) The system has unique solution if $k \neq 2, k \neq - 2 .$

(B) The system has unique solution if k = -2.

(C) The system has unique solution if k = 2.

(D) The system has no solution if k = 2.

(E) The system has infinite number of solutions if $k \neq - 2 .$

Which of the following statements are correct?

Vishal Baghel

x – 2y = 1, x – y + kz = -2, ky + 4z = 6

x – 2y + 0. z – 1 = 0

x – y + kz + 2 = 0

0x + ky + 4z – 6 = 0

$Δ_{1} = | \begin{matrix} 1 & - 2 & 0 \\ - 2 & - 1 & k \\ 6 & k & 4 \end{matrix} | = - (k + 10) (k + 2)$

For no solution

$Δ = 0, Δ_{1} \neq 0$

k = 2

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