SATHEE: Matrices and Determinants 2 Question 4

Matrices and Determinants 2 Question 4

4. If

$[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ $[\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$ $[\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}]$ $[\begin{matrix} 1 & n - 1 \\ 0 & 1 \end{matrix}]$ $= [\begin{matrix} 1 & 78 \\ 0 & 1 \end{matrix}]$ , then the inverse of $[\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}]$ is

(2019 Main, 9 April I)

(a) $[\begin{matrix} 1 & 0 \\ 12 & 1 \end{matrix}]$

(b) $[\begin{matrix} 1 & - 13 \\ 0 & 1 \end{matrix}]$

(d) $[\begin{matrix} 1 & - 12 \\ 0 & 1 \end{matrix}]$

Show Answer

Answer:

Correct Answer: 4. (b)

Solution:

Given

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

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

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

$= [\begin{matrix} 1 & (n - 1) + (n - 2) + \dots + 3 + 2 + 1 \\ 0 & 1 \end{matrix}]$

$= [\begin{matrix} 1 & \frac{n (n - 1)}{2} \\ 0 & 1 \end{matrix}]$ $= [\begin{matrix} 1 & 78 \\ 0 & 1 \end{matrix}]$

Since, both matrices are equal, so equating corresponding element, we get

$\begin{aligned} \frac{n (n - 1)}{2} & = 78 \Rightarrow n (n - 1) = 156 \\ = 13 \times 12 = 13 (13 - 1) \end{aligned}$

$\Rightarrow n = 13$

So, $A = [\begin{matrix} 1 & 13 \\ 0 & 1 \end{matrix}] = A^{- 1} = [\begin{matrix} 1 & - 13 \\ 0 & 1 \end{matrix}]$

$∵$ if $| A | = 1$ and $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $A^{- 1} = [\begin{matrix} d & - b \\ - c & a \end{matrix}]$

Matrices and Determinants 2 Question 4

4. If

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 2 Question 4

4. If

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu