If M is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct? S1: Each row of M can be represented as a linear combination of the other rows S2: Each column of M can be represented as a linear combination of the other columns S3: M X=0 has a nontrivial solution S4: M has an inverse

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Q69GATE 2008MCQ

The following system of equations $x_{1}+x_{2}+2x_{3}=1$ $x_{1}+2x_{2}+3x_{3}=2$ $x_{1}+4x_{2}+\alpha x_{3}=4$ has a unique solution. The only possible value(s) for $\alpha$ is/are

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Q70GATE 2008MCQ

If the two matrices

\begin{bmatrix} 1 &0 &x \\ 0 & x& 1\\ 0 & 1 & x \end{bmatrix} \text{and }\begin{bmatrix} x &1 &0 \\ x & 0& 1\\ 0 & x & 1 \end{bmatrix}

have the same determinant, then the value of $x$ is

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Q71GATE 2008MCQ

How many of the following matrices have an eigenvalue 1?

\begin{bmatrix} 1 & 0\\ 0&0 \end{bmatrix}\begin{bmatrix} 0 & 1\\ 0&0 \end{bmatrix}\begin{bmatrix} 1 & -1\\ 1&1 \end{bmatrix}

and

\begin{bmatrix} -1 &0 \\ 1&-1 \end{bmatrix}

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Q72GATE 2007MCQ

Eigen vectors of

\begin{bmatrix} 1 && \cos \theta \\ \cos \theta && 1 \end{bmatrix}

are

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Q73GATE 2007MCQ

Let A be the matrix

\begin{bmatrix}3 &1 \\ 1&2\end{bmatrix}

. What is the maximum value of $x^TAx$ where the maximum is taken over all x that are the unit eigenvectors of A?

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Q74GATE 2007MCQ

Let A be a 4 x 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of

\begin{bmatrix} A & I\\ I&A \end{bmatrix}

where I is the 4 x 4 identity matrix?

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📖 Explanation

Let $\lambda$ be an eigenvalue of matrix $A$ , and let $v$ be a corresponding eigenvector, so $Av = \lambda v$ .
Let the block matrix be

M = \begin{bmatrix} A & I\\ I&A \end{bmatrix}

. We seek its eigenvalues.
Consider a vector

X = \begin{bmatrix} v \\ v \end{bmatrix}

.
Then

M X = \begin{bmatrix} A & I\\ I&A \end{bmatrix} \begin{bmatrix} v \\ v \end{bmatrix} = \begin{bmatrix} Av + Iv \\ Iv + Av \end{bmatrix} = \begin{bmatrix} \lambda v + v \\ v + \lambda v \end{bmatrix} = \begin{bmatrix} (\lambda+1)v \\ (\lambda+1)v \end{bmatrix} = (\lambda+1) \begin{bmatrix} v \\ v \end{bmatrix}

.
Thus, $\lambda+1$ is an eigenvalue of $M$ .

Now consider a vector

Y = \begin{bmatrix} v \\ -v \end{bmatrix}

.
Then

M Y = \begin{bmatrix} A & I\\ I&A \end{bmatrix} \begin{bmatrix} v \\ -v \end{bmatrix} = \begin{bmatrix} Av - Iv \\ Iv - Av \end{bmatrix} = \begin{bmatrix} \lambda v - v \\ v - \lambda v \end{bmatrix} = \begin{bmatrix} (\lambda-1)v \\ -(\lambda-1)v \end{bmatrix} = (\lambda-1) \begin{bmatrix} v \\ -v \end{bmatrix}

.
Thus, $\lambda-1$ is an eigenvalue of $M$ .

The eigenvalues of $A$ are -5, -2, 1, 4.
The possible eigenvalues for $M$ are $\lambda \pm 1$ :
For $\lambda = -5$ : $-5 \pm 1 \implies -4, -6$ .
For $\lambda = -2$ : $-2 \pm 1 \implies -1, -3$ .
For $\lambda = 1$ : $1 \pm 1 \implies 0, 2$ .
For $\lambda = 4$ : $4 \pm 1 \implies 3, 5$ .

The set of all eigenvalues of $M$ is $\{-6, -4, -3, -1, 0, 2, 3, 5\}$ .
Comparing with the options, 2 is an eigenvalue of $M$ .

The final answer is $\boxed{C}$

Q75GATE 2006MCQ

Consider the polynomial $p(x)=a_{0}+a_{1}x+a_{2}x^{2} + a_{3}x^{3}$ , where $a_{i}\neq 0,\forall i$ . The minimum number of multiplications needed to evaluate p on an input x is:

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Q76GATE 2006MCQ

What are the eigenvalues of the matrix P given below

P= \begin{pmatrix} a &1 &0 \\ 1& a& 1\\ 0&1 &a \end{pmatrix}

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Q77GATE 2006MCQ

F is an nxn real matrix. b is an nx1 real vector. Suppose there are two nx1 vectors, u and v such that u $\neq$ v , and Fu=b, Fv=b. Which one of the following statements is false?

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Q78GATE 2005MCQ

What are the eigen values of the following 2 x 2 matrix?

\begin{bmatrix} 2 &-1 \\ -4 & 5 \end{bmatrix}

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Q79GATE 2005MCQ

Consider the following system of equations in three real variables $x_1, x_2 \; and \; x_3$ : $2x_1 - x_2 + 3x_3 = 1$ $3x_1 + 2x_2 + 5x_3 = 2$ $-x_1 + 4x_2 + x_3 = 3$ The system of equations has

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Q80GATE 2005MCQ

The determinant of the matrix given below is

\begin{bmatrix} 0 &1 &0 &2 \\ -1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& -2& 0& 1 \end{bmatrix}

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Linear Algebra Practice Questions

📘 Linear Algebra – Concept Summary