GATE-CH-1994-1-a-math-1mark

The inverse of a matrix \(\begin {pmatrix}a&0\\0&b\end {pmatrix}\) is

• \(\begin {pmatrix}ab&0\\0&1\end {pmatrix}\)

• \(\begin {pmatrix}b&0\\a&b\end {pmatrix}\)

• \(\begin {pmatrix}1/a&0\\0&1/b\end {pmatrix}\)

• \(\begin {pmatrix}a&0\\0&1/b\end {pmatrix}\)

GATE-CH-1995-1-a-math-1mark

The rank of matrix \(\begin {pmatrix}1&0&0\\0&2&0\\3&0&0\end {pmatrix}\)

• 0

• 1

• 2

• 3

GATE-CH-1999-1-1-math-1mark

The system of equations, \[ \begin {align*} 2x + 4y &= 10 \\
5x + 10y &= 25 \end {align*} \]

• has no unique solution

• has only one solution

• has only two solutions

• has infinite solutions

GATE-CH-1999-1-3-math-1mark

The rank of the matrix \(\begin {pmatrix}3&0&1&2\\4&7&3&3\\1&7&2&1\end {pmatrix}\) is

• 0

• 1

• 2

• 3

GATE-CH-2001-1-1-math-1mark

The value of the following determinant \[ \begin {vmatrix}1&0&0&0&0\\-2&2&0&0&0\\3&5&3&0&0\\-1&4&7&4&0\\-5&-6&2&1&1\end {vmatrix} \]

• 24

• 32

• -112

• 0

GATE-CH-2002-1-2-math-1mark

The inverse of the matrix \(\displaystyle \begin {pmatrix}0.2&0&0\\0&1&0\\0&0&0.5\end {pmatrix}\) is

• \(\displaystyle \begin {pmatrix}2&0&0\\0&1&0\\0&0&5\end {pmatrix}\)

• \(\displaystyle \begin {pmatrix} -0.2&0&0\\0&-1&0\\0&0&-0.5\end {pmatrix}\)

• \(\displaystyle \begin {pmatrix}5&0&0\\0&1&0\\0&0&2\end {pmatrix}\)

• \(\displaystyle \begin {pmatrix}0.5&0&0\\0&1&0\\0&0&0.2\end {pmatrix}\)

GATE-CH-2007-2-math-1mark

The value of “\(a\)” for which the following set of equations 

\[ \begin {align*} y+2z&= 0 \\ 2x+y+z&=0 \\ ax+2y&=0 \end {align*} \]

have non-trivial solution, is

• 0

• 8

• \(-2\)

• 3

GATE-CH-2009-3-math-1mark

A system of linear equations \(A\boldsymbol {x}=\boldsymbol {0}\). where \(A\) is an \(n\times n\) matrix, has a non-trivial solution ONLY if

• rank of \(A > n\)

• rank of \(A = n\)

• rank of \(A < n\)

• \(A\) is an identity matrix

GATE-CH-2010-11-math-1mark

The inverse of the matrix \(\displaystyle \begin {bmatrix}1&2\\3&4\end {bmatrix}\) is

• \(\displaystyle \begin {bmatrix}-2&-1\\-3/2&-1/2\end {bmatrix}\)

• \(\displaystyle \begin {bmatrix}-2&3/2\\1&-1/2\end {bmatrix}\)

• \(\displaystyle \begin {bmatrix}-2&1\\3/2&-1/2\end {bmatrix}\)

• \(\displaystyle \begin {bmatrix}2&-3/2\\-1&1/2\end {bmatrix}\)

GATE-CH-2012-1-math-1mark

Consider the following set of linear algebraic equations 

\[ \begin {align*} x_1+2x_2+3x_3&=2\\ x_2+x_3&=-1 \\ 2x_2+2x_3&=0 \end {align*} \]

 The system has

• a unique solution

• no solution

• an infinite number of solutions

• only the trivial solution

GATE-CH-2012-5-math-1mark

Consider the following \((2\times 2)\) matrix \[ \begin {pmatrix}4&0\\0&4\end {pmatrix}\] Which one of the following vectors is NOT a valid eigen vector of the above matrix ?

• \(\displaystyle \begin {pmatrix}1\\0\end {pmatrix}\)

• \(\displaystyle \begin {pmatrix}-2\\1\end {pmatrix}\)

• \(\displaystyle \begin {pmatrix}4\\-3\end {pmatrix}\)

• \(\displaystyle \begin {pmatrix}0\\0\end {pmatrix}\)

GATE-CH-2013-3-math-1mark

Which of the following statements are TRUE?

16. The eigenvalues of a symmetric matrix are real
17. The value of the determinant of an orthogonal matrix can only be \(+1\)
18. The transpose of a square matrix \(A\) has the same eigenvalues as those of \(A\)
19. The inverse of an ‘\(n \times n\)’ matrix exists if and only if the rank is less than ‘\(n\)’

• P and Q only

• P and R only

• Q and R only

• P and S only

GATE-CE-2008-1-math-1mark

The product of matrices \((PQ)^{-1}P\) is

• \(P^{-1}\)

• \(Q^{-1}\)

• \(P^{-1}Q^{-1}P\)

• \(PQP^{-1}\)

GATE-CE-2011-1-math-1mark

\([A]\) is a square matrix which is neither symmetric nor skew symmetric and \([A]^T\) is it’s transpose. The sum and difference of these matrices are defined as \([S] = [A] + [A]^T\) and \([D] = [A] - [A]^T \) respectively . Which of the following statements is TRUE?

• Both \([S]\) and \([D]\) are symmetric

• Both \([S]\) and \([D]\) are skew-symmetric

• \([S]\) is skew-symmetric and \([D]\) is symmetric

• \([S]\) is symmetric and \([D]\) is skew-symmetric

GATE-CH-EC-2008-1-math-1mark

All the four entries of \(2\times 2\) matrix \(P = \begin {pmatrix}p_{11}&p_{12}\\p_{21}&p_{22}\end {pmatrix}\) are non zero, and one of its eigen values is zero. Which of the following statements is true?

• \(p_{11}p_{22}-p_{12}p_{21}=1\)

• \(p_{11}p_{22}-p_{12}p_{21}=-1\)

• \(p_{11}p_{22}-p_{12}p_{21}=0\)

• \(p_{11}p_{22}+p_{12}p_{21}=0\)

GATE-CH-2015-2-math-1mark

For the matrix \(\displaystyle \begin {pmatrix}4 & 3 \\ 3 & 4 \end {pmatrix}\), if \(\displaystyle \begin {pmatrix}1 \\ 1 \end {pmatrix}\) is an eigenvector, then the corresponding eigenvalue is ____________

• Answer

GATE-CH-2000-2-2-math-2mark

The inverse of the matrix \(\begin {pmatrix} 1 & -1 \\ -1 & -1 \end {pmatrix}\)

• does not exist

• is \(\begin {pmatrix} 1 & 1 \\ 1 & -1 \end {pmatrix}\)

• is \(\begin {pmatrix} 0.5 & -0.5 \\ -0.5 & -0.5 \end {pmatrix}\)

• is \(\begin {pmatrix} -0.5 & 0.5 \\ 0.5 & 0.5 \end {pmatrix}\)

GATE-CH-2004-31-math-2mark

The sum of the eigen-values of the matrix \[ \begin {pmatrix} 3 & 4\\ x & 1 \end {pmatrix} \] for real and negative values of \(x\) is

• greater than zero

• less than zero

• zero

• dependent on the value \(x\)

GATE-CH-2004-32-math-2mark

The following system of equations has

\[ \begin {eqnarray*} 4x+6y &=& 8 \\ 7x+8y&=& 9 \\ 3x+2y &=& 1  \end {eqnarray*} \]

• no solution

• only one solution

• two solutions

• infinite number of solutions

GATE-CH-2005-31-math-2mark

How many solutions does the following system of equations have? \[ \begin {eqnarray*} 4x+2y+x&=&7 \\ x+3y+z&=&3 \\ 3x+4y+2z&=&2 \end {eqnarray*} \] 

• 0

• 1

• 2

• \(\infty \)

GATE-CH-2005-32-math-2mark

The matrix \(A\) is given by, \(\displaystyle A = \begin {pmatrix}1 & 4\\ a & 2\end {pmatrix}\). The eigen values of the matrix \(A\) are real and non-negative for the condition

• \((-1/16) \le a \le (1/16)\)

• \((-1/2) \le a \le (1/2)\)

• \((-1/2) \le a \le (1/16)\)

• \((-1/16) \le a \le (1/2)\)

GATE-CH-2006-21-math-2mark

If the following represents the equation of a line \[ \begin {vmatrix}x & 2 & 4\\ y & 8 & 0 \\ 1 & 1 & 1\end {vmatrix} = 0 \] then the line passes through the point

• \((0,0)\)

• \((3,4)\)

• \((4,3)\)

• \((4,4)\)

GATE-CH-2006-22-math-2mark

If \(\displaystyle A = \begin {bmatrix}2 & 1 \\ 2 & 3\end {bmatrix}\), then the eigen values of \(A^3\) are

• 27 and 8

• 64 and 1

• 12 and 3

• 4 and 1

GATE-CH-2007-22-math-2mark

\({A}\) and \({B}\) are two \(3\times 3\) matrix such that \[ A = \begin {bmatrix}-2 & 4 & 6\\ 1 & 2 & 1\\0 & 4 & 4\end {bmatrix}, \quad B = 0 \] and \(AB=0\). Then the rank of matrix \(B\) is

• \(r=2\)

• \(r<3\)

• \(r\le 3\)

• \(r=3\)

GATE-CH-EC-2012-47-math-2mark

If \(A = \begin {pmatrix}5&-3\\2&0\end {pmatrix}\) and \(I = \begin {pmatrix}1&0\\0&1\end {pmatrix}\), the value of \(A^3\) is

• \(15A+12I\)

• \(19A+30I\)

• \(17A+15I\)

• \(17A+21I\)

GATE-CH-2016-26-math-2mark

A set of simultaneous linear algebraic equations is represented in a matrix form as shown below. \[ \begin {bmatrix} 0 & 0 & 0 & 4 & 13 \\ 2 & 5 & 5 & 2 & 10 \\ 0 & 0 & 2 & 5 & 3 \\ 0 & 0 & 0 & 4 & 5 \\ 2 & 3 & 2 & 1 & 5 \end {bmatrix} \begin {bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end {bmatrix} = \begin {bmatrix} 46 \\ 161 \\ 61 \\ 30 \\ 81 \end {bmatrix} \] The value (rounded off to the nearest integer) of \(x_3\) is ____________

• Answer

GATE-CH-1994-5-math-5mark

Find the eigenvalues of the matrix \[ A = \begin {pmatrix}0&2\\-1&-1\end {pmatrix} \]

• Answer

GATE-CH-1996-9-math-5mark

Given the matrix \(A = \begin {bmatrix}-1&-2\\3&4\end {bmatrix}\)

1. Write down the characteristic equation
2. Compute \(A^4\) without direct multiplication.

• Answer

GATE-CH-1997-10-math-5mark

For the matrix \(A\) given below \[ A = \begin {bmatrix}2&0&0\\1&4&0\\3&5&6\end {bmatrix} \]

1. calculate its eigen values, and
2. determine the eigen vector corresponding to the lowest eigen value.

• Answer

GATE-CH-2002-5-math-5mark

Matrix \(\displaystyle A = \begin{pmatrix}0.1&0.5&0\\0.8&0&0.4\\0.1&0.5&0.6\end{pmatrix}\) has the property that it satisfies \(AX = X\), for any vector \(X\).

1. Write the characteristic equation to be solved for eigen values of \(A\).

2. Based on visual observation, find one of the eigen values of \(A\).

3. Find the other two eigen values of \(A\).

• Answer