If 1 2 A - A ' + I - 1 = 2 λ λ - 13 - λ 3 λ 3 - 17 10 - 1 7 - 11 5 for A = - 2 3 4 5 - 4 - 3 7 2 9 , then λ is:

A = - 2 3 4 5 - 4 - 3 7 2 9 So, A ' = - 2 5 7 3 - 4 2 4 - 3 9 Hence, S = 1 2 A - A ' + I = 1 2 × - 2 3 4 5 - 4 - 3 7 2 9 - - 2 5 7 3 - 4 2 4 - 3 9 + 1 0 0 0 1 0 0 0 1 = 1 2 - 1 - 3 2 1 1 2 - 5 2 3 2 5 2 1 2 Here, S = 1 2 1 4 + 25 4 - 1 - 1 2 + 15 4 + 3 2 5 2 + 3 4 = 39 8 A d j S = 13 2 - 13 4 13 4 - 17 4 5 2 - 1 4 7 4 - 11 4 5 4 So, S - 1 = 1 39 8 13 2 - 13 4 13 4 - 17 4 5 2 - 1 4 7 4 - 11 4 5 4 = 4 3 - 2 3 2 3 - 34 39 20 39 - 2 39 14 39 - 22 39 10 39 ⇒ 1 2 S - 1 = 1 39 26 - 13 13 - 17 10 - 1 7 - 11 5 So, λ = 39

Let a , b , c positive numbers. Prove that following system of equations in x , y and z x 2 a 2 + y 2 b 2 - z 2 c 2 = 1 ; x 2 a 2 - y 2 b 2 + z 2 c 2 = 1 ; - x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 has finitely many solutions

It is given that x 2 a 2 + y 2 b 2 - z 2 c 2 = 1 ; x 2 a 2 - y 2 b 2 + z 2 c 2 = 1 ; - x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Now let x 2 a 2 = X , y 2 b 2 = Y & z 2 c 2 = Z then the given system of equations can reduced to X + Y - Z = 1 X - Y + Z = 1 - X + Y + Z = 1 Now value of determinant of above system of equation will be A = 1 1 - 1 1 - 1 1 - 1 1 1 = 1 - 1 - 1 - 1 1 + 1 - 1 1 - 1 = A = - 4 ≠ 0 So the given system of equation have unique solution.

Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if

the product of entries in the main diagonal of M is not the square of an integer

the first column of M is the transpose of the second row of M

the second row of M is the transpose of first column of M

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 23

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IMPORTANT

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$A$ is a $(3 \times 3)$ diagonal matrix having integral entries such that $det (A) = 120,$ number of such matrices is $10 n .$ Then $n$ is:

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Important Questions on Matrices and Determinants

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 25

If

a_{1}, a_{2}, a_{3}, 5, 4, a_{6}, a_{7}, a_{8}, a_{9}

are in H.P. and

D = |\begin{matrix} a_{1} & a_{2} & a_{3} \\ 5 & 4 & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}|,

then the value of

21 D

is (Where

[.]

represents, the greatest integer function)

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 28

Suppose

A

is a matrix such that

A^{2} = A

and

(I + A)^{6} = I + k A,

then

k

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 32

The value of $a$ for which system of equation $a^{3} x + (a + 1)^{3} y + (a + 2)^{3} z = 0$ , $a x + (a + 1) y + (a + 2) z = 0$ and $x + y + z = 0$ has non-zero solutions is

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 34

The value of

' 2 k'

for which the set of equations

3 x + k y - 2 z = 0, x + k y + 3 z = 0, 2 x + 3 y - 4 z = 0

has a non-trivial solution over the set of rational is:

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 35

$A_{1} = [a_{1}]$

$A_{2} = [\begin{matrix} a_{2} & a_{3} \\ a_{4} & a_{5} \end{matrix}]$

$A_{3} = [\begin{array}{l} a_{6} & a_{7} & a_{8} \\ a_{9} & a_{10} & a_{11} \\ a_{12} & a_{13} & a_{14} \end{array}] \dots \dots \dots \dots \dots \dots \dots A_{n} = [\dots \dots .]$