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11th Maths Weekly Test -1( Matrices and Determinants )

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test -1( Matrices and Determinants )-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

Choose the correct statement

Matrix addition is not associative
Matrix addition is not commutative
Matrix multiplication is associative
Matrix multiplication is commutative

Choose the incorrect statement

Matrix multiplication is non commutative
Matrix addition is associative
Singular matrices have inverse
Non singular matrices have inverse

A root of the equation \(\begin{vmatrix} 3-x&-6 &3 \\ -6 & 3-x & 3 \\ 3 &3 &-6-x \end{vmatrix}=0 \ is\)

6
3
0
-6

The value of\(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1+sin\theta & 1 \\ 1 & 1 & 1+cos\theta \end{matrix} \right| \) is

3
1
2
\(\frac{1}{2}\)

The value of \(\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| \)=0, where a, b, c are in AP is

(x+1)(x+2)(x+3)
I
0
(x+a)(x+b)(x+c)

If \(\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix}=\begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}\), then the value of x+y is

5
6
4
3

If A(B+C)=AB + AC where A, B, C are matrices of the same order than the property applied is matrix multipication is

commutative
association
distributive over addition
distributive over multiplication

If a_ij=\({1\over2}(3i-2j)\) and A=[a_ij]_2x2 is

\(\begin{bmatrix} {1\over 2}& 2 \\ -{1\over2} & 1 \end{bmatrix}\)
\(\begin{bmatrix} {1\over 2}& -{1\over2} \\ 2& 1 \end{bmatrix}\)
\(\begin{bmatrix} 2& 2\\ {1\over 2}& -{1\over2} \end{bmatrix}\)
\(\begin{bmatrix} -{1\over 2}& {1\over2} \\ 1& 2 \end{bmatrix}\)

The negative of a matrix is obtained by multiplying it by

1
-1
I
A^T

Let A and B be two symmetric matrices of same order. Then which one of the following statement is not true?

A + B is a symmetric matrix
AB is a symmetric matrix
AB = (BA)^T
A^T B = AB^T

Part - B

Generic 2 - Answer All Question

Determine 3B + 4C - D if B, C, and D are given by
B= \(\begin{bmatrix} 2 & 3 & 0\\1 & -1 & 5\end{bmatrix}\) ,C=\(\begin{bmatrix} -1 & -2 & 3\\-1 & 0 & 2\end{bmatrix}\),D= \(\begin{bmatrix} 0 & 4 & -1\\5 & 6 & -5\end{bmatrix}\)
Construct an m × n matrix A= [a_ij], where a _ij is given by
\(a_{ij}={|3i-4j|\over 4}with \ m=3,n=4\)

Consider the matrix Aa=\(\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}\)
Find all possible real values of α satisfying the condition \(A\alpha +A^T_{\alpha}=I\)

Prove that \(\left[ \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right] =\left( a-b \right) \left( b-c \right) \left( c-a \right) \)

Compute |A| using Sarrus rule if A=\(\begin{bmatrix} 3& 4 & 1 \\ 0 &-1 &2 \\ 5 & -2 & 6 \end{bmatrix}\) .

Part - C

Generic 3 - Answer All Question

Construct a 2 × 3 matrix whose (i, j)^th element is given by \(a_ij={\sqrt{3}\over 2}|2i-3j|(1\le i\le2,1\le j\le3)\) .
Determine the values of a so that the following matrices are singular: A=\(\begin{bmatrix} 7& 3 \\ -2 & a \end{bmatrix}\)

Show that f(x)f(y)=f(x+y) where f(x)=\(\begin{bmatrix} cos \ x & -sin \ x & 0 \\ sin x & cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}\).

Prove that \(\left| \begin{matrix} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{matrix} \right| =2\left( a+b \right) \left( b+c \right) \left( c+a \right) \)

Using factor method show that \(\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \) = (a - b) (b - c) (c - a).

Part - D

Generic 5 - Answer All Question

If \(A=\left[ \begin{matrix} 1 & 8 \\ 4 & 3 \end{matrix} \right] \quad B=\left[ \begin{matrix} 1 & 3 \\ 7 & 4 \end{matrix} \right] \quad C=\left[ \begin{matrix} -4 & 6 \\ 3 & -5 \end{matrix} \right] \)
Prove that
(l) AB ≠ BA
(it)A(BC) = (AB)C
(iii) A(B + C) = AB + AC
(iv) AI = IA = A

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} 2& 3 &4 \\ 5 & 6 & 8 \\ 6x & 9x &12x \end{vmatrix}\)

If \(\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the values of p,q, and r.

If A and B are symmetric matrices of same order, prove that
AB + BA is a symmetric matrix

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