# Linear Algebra

 Question 1
Cholesky decomposition is carried out on the following square matrix $[\mathrm{A}]$.

$[A]=\left[\begin{array}{cc} 8 & -5 \\ -5 & a_{22} \end{array}\right]$

Let $\mathrm{I}_{\mathrm{ij}}$ and $\mathrm{a}_{\mathrm{ij}}$ be the $(i, j)^{\text {th }}$ elements of matrices $[L]$ and $[A]$, respectively. If the element $I_{22}$ of the decomposed lower triangular matrix $[\mathrm{L}]$ is 1.968 , what is the value (rounded off to the nearest integer) of the element $a_{22}$ ?
 A 5 B 7 C 9 D 11
GATE CE 2023 SET-2   Engineering Mathematics
Question 1 Explanation:
We know, cholesky decomposition,
$A=LL^{T}$
Where, $L=$ lower tringular matrix
$\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]=\left[\begin{array}{cc}L_{11} & 0 \\ L_{21} & L_{22}\end{array}\right]\left[\begin{array}{cc}\mathrm{L}_{11} & L_{21} \\ 0 & L_{22}\end{array}\right]$

$\left[\begin{array}{cc}8 & -5 \\ -5 & \mathrm{a}_{22}\end{array}\right]=\left[\begin{array}{cc}\mathrm{L}_{11}^{2} & \mathrm{~L}_{11} \mathrm{~L}_{21} \\ \mathrm{~L}_{11} \mathrm{~L}_{21} & \mathrm{~L}_{21}^{2}+\mathrm{L}_{22}^{2}\end{array}\right]$
On comparison on both sides,
$\mathrm{L}_{11}=\sqrt{8}=2 \sqrt{2}$
$and,\;\; L_{11} L_{21}=-5$
$\mathrm{L}_{21}=\frac{-5}{2 \sqrt{2}}$
$and,\;\; a_{22}=L_{21}^{2}+L_{22}^{2}$
$=\left(\frac{-5}{2 \sqrt{2}}\right)^{2}+1.968^{2}$
$=6.998 \approx 7$
 Question 2
For the matrix
$[A]=\left[\begin{array}{ccc} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{array}\right]$
which of the following statements is/are TRUE?
 A $[A]\{X\}=\{b\}$ has a unique solution B $[A]\{X\}=\{b\}$ does not have a unique solution C $[A]$ has three linearly independent eigenvectors D $[\mathrm{A}]$ is a positive definite matrix
GATE CE 2023 SET-2   Engineering Mathematics
Question 2 Explanation:
$|A|=\left|\begin{array}{ccc}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right|$
$=1(2-1)+1(-1)+0$
$=1-1=0$

So $A X=B$ does not have unique solution because $\rho(A) \lt 3$
$|A-\lambda|=0 \Rightarrow\left|\begin{array}{ccc}1-\lambda & -1 & 0 \\ -1 & 2-\lambda & -1 \\ 0 & -1 & 1-\lambda\end{array}\right|=0$
$(1-\lambda)((2-\lambda)(1-\lambda)-1)+1(-1+\lambda)+0=0$
$(1-\lambda)\left(\lambda^{2}-3 \lambda+1\right)-1+\lambda=0$
$(1-\lambda)\left(\lambda^{2}-3 \lambda+1-1\right)=0$
$(\lambda-1)\left(\lambda^{2}-3 \lambda\right)=0$
$\lambda=0,1,3$

Matrix $A$ has three distinct Eigen values so have three linearly independent eigen vectors. so option (C) is correct.
Given matrix is symmetric matrix with real value entries. Hence $A$ is not a positive definite matrix. because
$|1|=1 \gt 0$
$\left|\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right|=2-1=1>0$
$\left|\begin{array}{ccc}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{array}\right|=0$ (which is not positive)
Hence option (D) is incorrect.

 Question 3
For the matrix
$[A]=\left[\begin{array}{lll} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{array}\right]$
Which of the following statements is/are TRUE?
 A The eigenvalues of $[A]^T$ are same as the eigenvalues of $[A]$ B The eigenvalues of $[A]^{-1}$are the reciprocals of the eigenvalues of $[A]$ C The eigenvectors of $[A]^T$ are same as the eigenvectors of $[A]$ D The eigenvectors of $[A]^{-1}$ are same as the eigenvectors of $[A]$
GATE CE 2023 SET-1   Engineering Mathematics
Question 3 Explanation:
$A x=\lambda x \ldots$ (i)
$A^{T} x=\lambda x \ldots$ (ii)
$A$ and $A^T$ both have same eigen values and eigen vectors.
$A x=\lambda x \ldots(i)$
$\Rightarrow \quad A^{-1} A x=A^{-1}(\lambda x)=\lambda A^{-1} x$
$\Rightarrow \quad x=\lambda A^{-1} x$
$A^{-1} x=\frac{1}{\lambda} x$
So, eigen value and eigen vector of $A^{-1}$ is $\frac{1}{\lambda} x$ and $\mathrm{x}$.
 Question 4
If $M$ is an arbitrary real $n \times n$ matrix, then which of the following matrices will have non-negative eigenvalues?
 A $M^2$ B $MM^T$ C $M^TM$ D $(M^T)^2$
GATE CE 2023 SET-1   Engineering Mathematics
Question 4 Explanation:
$M X=\lambda X$
$[\lambda$ is eigen value of $(M)]$
\begin{aligned} M M X & =\lambda M X \\ M^{2} X & =\lambda(\lambda X) \\ M^{2} X & =\lambda^{2} X \end{aligned}

$\left[\lambda^{2}\right.$ (non negative) is eigen value of $\left.\mathrm{M}^{2}\right]$
\begin{aligned} M X & =\lambda X \\ M^{\top} X & =\lambda X \end{aligned}

$\left[M, M^{\top}\right.$ have same eigen values]
\begin{aligned} & M M^{\top} X=\lambda M X \\ & M M^{\top} X=\lambda(\lambda X) \\ & M M^{\top} X=\lambda^{2} X \end{aligned}

$\left[\lambda^{2}\right.$ (none negative) is eigen value of $\left.M M^{\top}\right]$
\begin{aligned} M X & =\lambda X \\ M^{\top} X & =\lambda X \\ M^{\top} M X & =\lambda M X \\ M^{\top} M X & =\lambda(\lambda X) \\ M^{\top} M X & =\lambda^{2} X \end{aligned}

$\left[\lambda^{2}\right.$ (non negative) is eigen value of $\left.M^{\top} M\right]$
\begin{aligned} M X & =\lambda X \\ M M X & =\lambda M X \\ M^{2} X & =\lambda(\lambda X) (M^T)^2 X&= \lambda ^2 X \end{aligned}
$\left[\because M, M^{\top}\right.$ have same eigen value]
$\left(M^{\top}\right)^{2} X=\lambda^{2} X$
[ $\lambda^{2}$ is eigen value of $\left(M^{\top}\right)^{2}$ which non negative] Hence, option A, B, C, D are correct.
 Question 5
Let $y$ be a non-zero vector of size 2022 x 1. Which of the following statement(s) is/are TRUE?
 A $yy^T$ is a symmetric matrix. B $y^Ty$ is an eigenvalue of $yy^T$ C $yy^T$ has a rank of 2022. D $yy^T$ is invertible.
GATE CE 2022 SET-2   Engineering Mathematics
Question 5 Explanation:
Let vector
\begin{aligned} y&=\begin{bmatrix} 4\\ 4\\ 4 \end{bmatrix}_{3 \times 1}\\ y^T&=\begin{bmatrix} 4& 4& 4 \end{bmatrix}_{1 \times 3}\\ yy^T&=\begin{bmatrix} 4\\ 4\\ 4 \end{bmatrix}\begin{bmatrix} 4& 4& 4 \end{bmatrix}\\ &=\begin{bmatrix} 16 & 16 & 16\\ 16 & 16 & 16\\ 16 & 16 & 16 \end{bmatrix}\\ y^Ty&=[4^2+4^2+4^2]_{1 \times 1}\\ &=[48]_{1 \times 1}\\ \rho (y)&=\rho (y^T)=\rho (yy^T)=\rho (y^Ty)=1 \end{aligned}
From above information
$yy^T$ is asymmetric.
$y^Ty$ is an eigen value of $yy^T$.
$yy^T$ has rank 1
$det(yy^T) =0$ so, $yy^T$ is not invertible.

There are 5 questions to complete.

### 1 thought on “Linear Algebra”

1. Question no 11 correct matrix option has some values incorrect