Mathematics

Linear Algebra

449 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice general knowledge

Matrix 1

  1. 1999

  2. 1998

  3. 2000

  4. 1997
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Matrix film was released in 1999, making option A correct. The other years (1997, 1998, 2000) are close but incorrect for this landmark sci-fi film.

Multiple choice technology

A Determinant is a

  1. Used for group by and to avoid double counting of data

  2. Join to prevent loops

  3. Function in Framework manager

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A determinant in Framework Manager defines a set of attributes that uniquely determine other attributes, preventing double-counting in query results. When you group by the determinant, Framework Manager knows how to handle the dependent columns correctly. Options B and C are wrong - determinants are not joins or functions; they're metadata about data dependencies that guide query generation.

Multiple choice

$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Which one of the following statements is false?

  1. Determinant of F is zero

  2. There are an infinite number of solutions to Fx = b
  3. There is an $x≠0$ such that $Fx = 0$

  4. F must have two identical rows
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(A) : Correct. We are given

$$Fu = b$$

$$Fv = b$$

So $F(u-v) = 0$

Since $u \neq v$, so we have a non-zero solution $w = (u-v)$ to homogeneous equation $Fx=0$. Now any vector $\lambda w$ is also a solution of $Fx=0$, and so we have infinitely many solutions of $Fx=0$, and so determinant of F is zero.

(B) : Correct. Consider a vector $u+\lambda w$.

$$F(u+\lambda w) = Fu+F(\lambda w) = b + 0 = b$$

So there are infinitely many vectors of the form $u+\lambda w$, which are solutions to equation $Fx=b$.

(C) : Correct. In option (a), we proved that vector $(u-v) \neq 0$ satisfies equation $Fx=0$.

(D) : False. This is not necessary.

So option (D) is the answer.

Multiple choice

Let A be a $4 \times 4$ matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix?

  1. -5

  2. -7

  3. 2

  4. 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The block matrix [A I; I A] has eigenvalues that are eigenvalues of A+I and A-I (for eigenvectors where the top and bottom halves are aligned or anti-aligned). For eigenvalue -5 of A: -5+1=-4 and -5-1=-6. For -2: -2+1=-1, -2-1=-3. For 1: 1+1=2, 1-1=0. For 4: 4+1=5, 4-1=3. Therefore 1 is an eigenvalue (from eigenvalue 0).

Multiple choice

How many of the following matrices have an eigenvalue 1? $\left[\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array} \right]\left[\begin{array}{cc}0 & 1 \\ 0 & 0 \end{array} \right] \left[\begin{array}{cc}1 & -1 \\ 1 & 1 \end{array} \right]$ and $\left[\begin{array}{cc}-1 & 0 \\ 1 & -1 \end{array} \right]$

  1. one

  2. two

  3. three

  4. four
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

x : eigen value x2 - trace (A) + det (A) = 0, $\lambda$= 1 $\left[\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array} \right]$: 1-2 + 1 = 0 $\left[\begin{array}{cc}0 & 1 \\ 0 & 0 \end{array} \right]: 1-0 + 0 \ne 0 $ $ \left[\begin{array}{cc}1 & -1 \\ 1 & 1 \end{array} \right]: 1-2 + 2 \ne 0$ $\left[\begin{array}{cc}-1 & 0 \\ 1 & -1 \end{array} \right]: 1+2 + 1\ne 0 $

Multiple choice

Four matrices M1, M2, M3, and M4 have dimensions p x q, q x r, r x s, and s x t respectively can be multiplied in several ways with different number of total scalar multiplications. For example, when multiplied as ((M1 x M2) x (M3 x M4)) total number of scalar multiplications is pqr + rst + prt.

When multiplied as (((Mi x M2) x M3) x M4) the total number of scalar multiplications is pqr + prs + pst.

If p = 10, q = 100, r = 20, s = 5, and t = 80, then what is the minimum number of scalar multiplications needed ?

  1. 248000

  2. 44000

  3. 19000

  4. 25000
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Multiply as $(M_3 \times M_2 \times M_3) \times M_4$ The total number of scalar multiplication is = qrs + pqs + pst = 10000 + 5000 + 4000 = 19000

Multiple choice

Which one of the following does NOT equal $\begin{vmatrix} 1& x & x^{2}\\ 1& y & y^{2}\\ 1& z & z^{2} \end{vmatrix}?$
  1. $\begin{vmatrix} 1& x(x+1) & x+1\\\\ 1& y(y+1) & y+1\\\\ 1& z(z+1) & z+1 \end{vmatrix}$
  2. $\begin{vmatrix} 1& x+1 & x^{2}+1\\\\ 1& y+1 & y^{2}+1\\\\ 1& z+1 & z^{2}+1 \end{vmatrix}$
  3. $\begin{vmatrix} 0& x-y & x^{2}-y^{2}\\\\ 0& y-z & y^{2}-z^{2}\\\\ 1& z & z^{2} \end{vmatrix} $
  4. $\begin{vmatrix} 2& x+y & x^{2}+y^{2}\\\\ 2& y+z & y^{2}+z^{2}\\\\ 1& z & z^{2} \end{vmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If matrix B is obtained from matrix A by replacing the lth row by itself plus k times the mth row for l $\sigma$ m, then det(2) = det(1). With this property, given matrix is equal to the matrices given in the options (2), (3) and (4).

Multiple choice

Given A= $ \left[ \begin{array} \ 1 & 0 \\ 0 & 1 \end{array} \right] $ the state transition matrix eAt is given by
  1. $ \left[ \begin{array} \ 0 & e^{-t} \\\\ e^{-t} & 0 \end{array} \right] $
  2. $ \left[ \begin{array} \ e^{-t} & 0 \\\\ 0 & e^t & \end{array} \right] $
  3. $ \left[ \begin{array} \ e^{-t} & 0 \\\\ 0 & e^{-t} & \end{array} \right] $
  4. $ \left[ \begin{array} \ 0 & e^{t} \\\\ e^{t} & 0 \end{array} \right] $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation