1. Find the rank of the matrix A=\(\begin{bmatrix}
1 & 3 & 5\\
4 & 6 & 7\\
1 & 2 & 2\\
\end{bmatrix}\).
a) 3
b) 2
c) 1
d) 0
Explanation:To find out the rank of the matrix first find the |A|
If the value of the|A| = 0 then the matrix is said to be reduced
But, as the determinant of A has some finite value then, the rank of the matrix is 3.
2. The rank of the matrix (m × n) where m<n cannot be more than?
a) m
b) n
c) m*n
d) m-n
Explanation:let us consider a 2×3 matrix \(\begin{bmatrix}
1 & 1 & 1\\
4 & 5 & 6\\
\end{bmatrix}\)
Where R1≠R2 rank is 2
Another 2×3 matrix \(\begin{bmatrix}
1 & 1 & 1\\
1 & 1 & 1\\
\end{bmatrix}\)
Here, R1=R2 rank is 1
And the rank of these two matrices is 1, 2
So rank is cannot be more than m.
3. Given A=\(\begin{bmatrix}
2 & -0.1 \\
0 & 3 \\
\end{bmatrix} A^{-1} = \begin{bmatrix}
1/2 & a \\
0 & b \\
\end{bmatrix}\) then find a + b.
a) \(\frac{6}{20}\)
b) \(\frac{7}{20}\)
c) \(\frac{8}{20}\)
d) \(\frac{5}{20}\)
Explanation: : AA-1 = I = \(\begin{bmatrix}
1 & 2-0.1b \\
0 & 3b \\
\end{bmatrix}=\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}\)
Therefore, a = \(\frac{1}{60}\) and b = \(\frac{1}{3}\) and a + b = \(\frac{7}{20}\).
4. If a square matrix B is skew symmetric then.
a) BT = -B
b) BT = B
c) B-1 = B
d) B-1 = BT
Explanation:The transpose of a skew symmetric matrix should be equal to the negative of the matrix
Example: let us consider a matrix B = \(\begin{bmatrix}
a & e & d\\
-e & b & f\\
-d & -f & c\\
\end{bmatrix}\), BT = \(\begin{bmatrix}
a & -e & -d\\
e & b & -f\\
d & f & c\\
\end{bmatrix}\).
5. For the following set of simultaneous equations 1.5x-0.5y=2, 4x+2y+3z=9, 7x+y+5=10.
a) The solution is unique
b) Infinitely many solutions exist
c) The equations are incompatible
d) Finite number of multiple solutions exist
Explanation: The equations can be written as \(\begin{bmatrix}
1.5 & -0.5 & 0\\
4 & 2 & 3\\
7 & 1 & 5\\
\end{bmatrix}\)
It can also be written as A = \(\begin{bmatrix}
3 & -2 & 0\\
4 & 2 & 3\\
7 & 1 & 5\\
\end{bmatrix}\), |A|=19
Hence, it has a unique solution
6. Find the Eigen values of matrix \(A = \begin{bmatrix}
2 & 1 & 0\\
1 & 2 & 1\\
0 & 1 & 2\\
\end{bmatrix}\).
a) 2 + \(\sqrt{2}\), 2-\(\sqrt{2}\), 2
b) 2, 1, 2
c) 2, 2, 0
d) 2, 2, 2
Explanation: To find the Eigen values it satisfy the condition, |A-λI|=0
|A-λI| = \(\begin{bmatrix}
2 & 1 & 0\\
1 & 2 & 1\\
0 & 1 & 2\\
\end{bmatrix} – λ\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{bmatrix}\)
|A-λI| = \(\begin{vmatrix}
2-λ & 1 & 0\\
1 & 2-λ & 1\\
0 & 1 & 2-λ\\
\end{vmatrix}\)
= 2 – (λ2-4λ+3) – (2-λ)
By solving the above equation, we get,
λ = 2 + \(\sqrt{2}\), 2-\(\sqrt{2}\), 2.
7.Find the product of Eigen values of a matrix \(A = \begin{bmatrix}
1 & 2 & 4\\
0 & 6 & 0\\
3 & 1 & 2\\
\end{bmatrix}\).
a) 60
b) 45
c) -60
d) 40
Explanation: According to the property of Eigen values, the product of the Eigen values of a given matrix is equal to the determinant of the matrix |A| = 1(12-0) – 2(0) + 4(8)
= -60
8. Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.
a) a, b, c
b) -a, -b, -c
c) a-b, b-a, c-a
d) a-1, b-1, c-1
Explanation: According to the property of the Eigen values, any square matrix A and its transpose AT have the same Eigen values
9. What is Eigen value?
a) A vector obtained from the coordinates
b) A matrix determined from the algebraic equations
c) A scalar associated with a given linear transformation
d) It is the inverse of the transform
Explanation: Eigen values is a scalar associated with a given linear transformation of a vector space and having the property that there is some nonzero vector which is when multiplied by the scalar is equal to the vector obtained by letting the transformation operate on the vector
10. Find the sum of the Eigen values of the matrix \(A = \begin{bmatrix}
3 & 6 & 7\\
5 & 4 & 2\\
7 & 9 & 1\\
\end{bmatrix}\).
a) 7
b) 8
c) 9
d) 10
Explanation: According to the property of the Eigen values, the sum of the Eigen values of a matrix is its trace that is the sum of the elements of the principal diagonal.
Therefore, the sum of the Eigen values = 3 + 4 + 1 = 8.