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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Matrices

NDA 2023 · Mathematics · Matrices

If A=[ array l 1 2 3 array ] , then what is value of (I+A A^ ) ,...

If A=\left[123\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], then what is value of det⁡⁡(I+AA′)\operatorname\det \left(I+A A^{\prime}\right), where I is the 3×3identitymatrix?3 \times3 identity matrix?

Options

  1. A15
  2. B6
  3. C0
  4. D-1

Correct answer

A. 15

Step-by-step solution

Since, I + AA' =[100010001]+[123][123]=[100010001]+[123246369]=[2232563610]\begin{aligned} & =\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]+\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]\left[\begin{array}{lll} 1 & 2 & 3 \end{array}\right] \\ & =\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]+\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{array}\right]=\left[\begin{array}{ccc} 2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 6 & 10 \end{array}\right] \end{aligned} Now, \left|\mathrm{I}+\mathrm{AA}^{\prime}\right|=2(50-36)-2(20-18)+3(12-15)=15
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