Question 1 |

Which of the following is/are the eigenvector(s) for the matrix given below?

\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}

\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}

**MSQ**\begin{pmatrix}
-1\\
1\\
0\\
1
\end{pmatrix} | |

\begin{pmatrix}
1\\
0\\
-1\\
0
\end{pmatrix} | |

\begin{pmatrix}
-1\\
0\\
0\\
0
\end{pmatrix} | |

\begin{pmatrix}
0\\
1\\
-3\\
0
\end{pmatrix} |

Question 1 Explanation:

Question 2 |

Consider solving the following system of simultaneous equations using LU decomposition.

\begin{aligned} x_1+x_2-2x_3&=4 \\ x_1+3x_2-x_3&=7 \\ 2x_1+x_2-5x_3&=7 \end{aligned}

where L and U are denoted as

L= \begin{bmatrix} L_{11} & 0 & 0 \\ L_{21}& L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{bmatrix}, U= \begin{bmatrix} U_{11} & U_{12} & U_{13} \\ 0& U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{bmatrix}

Which one of the following is the correct combination of values for L32, U33, and x_1?

\begin{aligned} x_1+x_2-2x_3&=4 \\ x_1+3x_2-x_3&=7 \\ 2x_1+x_2-5x_3&=7 \end{aligned}

where L and U are denoted as

L= \begin{bmatrix} L_{11} & 0 & 0 \\ L_{21}& L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{bmatrix}, U= \begin{bmatrix} U_{11} & U_{12} & U_{13} \\ 0& U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{bmatrix}

Which one of the following is the correct combination of values for L32, U33, and x_1?

L_{32}=2,U_{33}=-\frac{1}{2},x_1=-1 | |

L_{32}=2,U_{33}=2,x_1=-1 | |

L_{32}=-\frac{1}{2},U_{33}=2,x_1=0 | |

L_{32}=-\frac{1}{2},U_{33}=-\frac{1}{2},x_1=0 |

Question 2 Explanation:

Question 3 |

Consider the following two statements with respect to the matrices A_{m \times n},B_{n \times m},C_{n \times n} \text{ and }D_{n \times n},

Statement 1: tr(AB) = tr(BA)

Statement 2: tr(CD) = tr(DC)

wheretr() represents the trace of a matrix. Which one of the following holds?

Statement 1: tr(AB) = tr(BA)

Statement 2: tr(CD) = tr(DC)

wheretr() represents the trace of a matrix. Which one of the following holds?

Statement 1 is correct and Statement 2 is wrong. | |

Statement 1 is wrong and Statement 2 is correct. | |

Both Statement 1 and Statement 2 are correct. | |

Both Statement 1 and Statement 2 are wrong. |

Question 3 Explanation:

Question 4 |

Suppose that P is a 4x5 matrix such that every solution of the equation Px=0 is a scalar multiple of \begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T. The rank of P is _______

1 | |

2 | |

3 | |

4 |

Question 4 Explanation:

Question 5 |

Consider the following matrix.

\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}

The largest eigenvalue of the above matrix is __________.

\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}

The largest eigenvalue of the above matrix is __________.

1 | |

3 | |

4 | |

6 |

Question 5 Explanation:

Question 6 |

If x+2 y=30,then \left(\frac{2 y}{5}+\frac{x}{3}\right)+\left(\frac{x}{5}+\frac{2 y}{3}\right) will be equal to

8 | |

16 | |

18 | |

20 |

Question 6 Explanation:

Question 7 |

Let A and B be two nxn matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB)=rank (A)rank (B)

II. det(AB)=det(A)det(B)

III. rank(A+B) \leq rank (A) + rank (B)

IV. det(A+B) \leq det(A) + det(B)

Which of the above statements are TRUE?

I. rank(AB)=rank (A)rank (B)

II. det(AB)=det(A)det(B)

III. rank(A+B) \leq rank (A) + rank (B)

IV. det(A+B) \leq det(A) + det(B)

Which of the above statements are TRUE?

I and II only | |

I and IV only | |

II and III only | |

III and IV only |

Question 7 Explanation:

Question 8 |

Consider the following matrix:

\begin{bmatrix} 1 & 2 & 4 & 8\\ 1& 3 & 9 &27 \\ 1 & 4 & 16 &64 \\ 1 & 5 & 25 &125 \end{bmatrix}

The absolute value of the product of Eigenvalues of R is _________ .

\begin{bmatrix} 1 & 2 & 4 & 8\\ 1& 3 & 9 &27 \\ 1 & 4 & 16 &64 \\ 1 & 5 & 25 &125 \end{bmatrix}

The absolute value of the product of Eigenvalues of R is _________ .

10 | |

12 | |

25 | |

125 |

Question 8 Explanation:

Question 9 |

Let X be a square matrix. Consider the following two statements on X.

I. X is invertible

II. Determinant of X is non-zero

Which one of the following is TRUE?

I. X is invertible

II. Determinant of X is non-zero

Which one of the following is TRUE?

I implies II; II does not imply I | |

II implies I; I does not imply II | |

I does not imply II; II does not imply I | |

I and II are equivalent statements |

Question 9 Explanation:

Question 10 |

Consider a matrix P whose only eigenvectors are the multiples of \begin{bmatrix} 1\\ 4 \end{bmatrix}.

Consider the following statements.

(I) P does not have an inverse

(II) P has a repeated eigenvalue

(III) P cannot be diagonalized

Which one of the following options is correct?

Consider the following statements.

(I) P does not have an inverse

(II) P has a repeated eigenvalue

(III) P cannot be diagonalized

Which one of the following options is correct?

Only I and III are necessarily true | |

Only II is necessarily true | |

Only I and II are necessarily true | |

Only II and III are necessarily true |

Question 10 Explanation:

There are 10 questions to complete.

Please make a login portal, so we can track our progress

Dear Chinmay Rajpurohit,

We will consider your suggestion.

Agree

Agreed

Hey Team,

Linear Algebra

In Q1 from Gate 2022 Option C should be [-1 0 2 2]ᵀ