if a beginarray20c 1 and 0 1 and 7 endarray and i

If \[{A} = \left[ {\begin{array}{*{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right]\] and \[{I} = \left[ {\begin{array}{*{20}{c}} 1&0\\ { 0}&1 \end{array}} \right],\] then the value of $$k$$ so that $$A^2 = 8A + kI$$ is

A. $$k = 7$$

B. $$k = - 7$$

C. $$k = 0$$

D. None of these

Answer : $$k = - 7$$

Solution :
\[\begin{array}{l} {A^2} = \left[ {\begin{array}{*{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right]\,\left[ {\begin{array}{*{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0\\ { - 8}&{49} \end{array}} \right]\\ {\rm{and}}\,\,\,8A + kI = 8\left[ {\begin{array}{*{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right] + k\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]\\ = \,\left[ {\begin{array}{*{20}{c}} 8&0\\ { - 8}&{56} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} k&0\\ 0&k \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {8 + k}&0\\ { - 8}&{56 + k} \end{array}} \right]\,\,{\rm{Thus}},\,\,{A^2} = 8A + kI\\ \Rightarrow \,\left[ {\begin{array}{*{20}{c}} 1&0\\ { - 8}&{49} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {8 + k}&0\\ { - 8}&{56 + k} \end{array}} \right]\\ \Rightarrow \,1 = 8 + k\,\,{\rm{and}}\,\,56 + k = 49\\ \Rightarrow \,k = - 7 \end{array}\]

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty

B. $$B$$ has as many elements as $$C$$

C. $$A = B \cup C$$

D. $$B$$ has twice as many elements as elements as $$C$$

View Answer

Releted Question 2

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0

B. 1

C. $$i$$

D. $$\omega $$

View Answer

Releted Question 3

Let $$a, b, c$$ be the real numbers. Then following system of equations in $$x, y$$ and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$ has

A. no solution

B. unique solution

C. infinitely many solutions

D. finitely many solutions

View Answer

Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$

B. $$A + B = A - B$$

C. $$A - B = B - A$$

D. $$AB=BA$$

View Answer

If \[{A} = \left[ {\begin{array}{{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right]\] and \[{I} = \left[ {\begin{array}{{20}{c}} 1&0\\ { 0}&1 \end{array}} \right],\] then the value of $$k$$ so that $$A^2 = 8A + kI$$ is

Releted MCQ Question on
Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Matrices and Determinants

Practice More MCQ Question on Maths Section

If \[{A} = \left[ {\begin{array}{*{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right]\] and \[{I} = \left[ {\begin{array}{*{20}{c}} 1&0\\ { 0}&1 \end{array}} \right],\] then the value of $$k$$ so that $$A^2 = 8A + kI$$ is

Releted MCQ Question on Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then \[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Practice More MCQ Question on Maths Section

If \[{A} = \left[ {\begin{array}{{20}{c}} 1&0\\ { - 1}&7 \end{array}} \right]\] and \[{I} = \left[ {\begin{array}{{20}{c}} 1&0\\ { 0}&1 \end{array}} \right],\] then the value of $$k$$ so that $$A^2 = 8A + kI$$ is

Releted MCQ Question on
Algebra >> Matrices and Determinants

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

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Matrices and Determinants