Linear Algebra - Study Mode

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[#61] Let A be the 2 × 2 matrix with elements a 11 = a 12 = a 21 = +1 and a 22 = -1. Then the eigen values of the matrix A 19 are

(A) 1024 and -1024

(B) 1024√2 and -1024√2

(C) 4√2 and -4√2

(D) 512√2 and -512√2

Correct Answer

(D) 512√2 and -512√2

[#62] How many of the following matrices have an eigen value 1? [left[ {x08egin{array}{{20}{c}}
1&0 \
0&0
end{array}}
ight],left[ {x08egin{array}{{20}{c}}
0&1 \
0&0
end{array}}
ight],left[ {x08egin{array}{{20}{c}}
1&{ - 1} \
1&1
end{array}}
ight]{ ext{ and }}left[ {x08egin{array}{{20}{c}}
{ - 1}&0 \
1&{ - 1}
end{array}}
ight]]

(A) one

(B) two

(C) three

(D) four

Correct Answer

(A) one

[#63] A real square matrix A is called skew-symmetric if

(A) A T = A

(B) A T = A -1

(C) A T = -A

(D) A T = A + A -1

Correct Answer

(C) A T = -A

[#64] If [{ ext{R}} = left[ {x08egin{array}{*{20}{c}}
1&0&{ - 1} \
2&1&{ - 1} \
2&3&2
end{array}}
ight],] xa0 xa0then top row of R -1 is

(A) [5 6 4]

(B) [5 -3 1]

(C) [2 0 -1]

(D) [left[ {x08egin{array}{*{20}{c}} 2&{ - 1}&{frac{1}{2}} end{array}} ight]]

Correct Answer

(B) [5 -3 1]

[#65] Let, [{ ext{A}} = left[ {x08egin{array}{{20}{c}}
2&{ - 0.1} \
0&3
end{array}}
ight]] xa0 and [{{ ext{A}}^{ - 1}} = left[ {x08egin{array}{{20}{c}}
{frac{1}{2}}&{ ext{a}} \
0&{ ext{b}}
end{array}}
ight].] Then (a + b) = ?

(A) [frac{7}{{20}}]

(B) [frac{3}{{20}}]

(C) [frac{{19}}{{60}}]

(D) [frac{{11}}{{20}}]

Correct Answer

(A) [frac{7}{{20}}]