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Chapter 4: Problem 292

If $$ A=\left|\begin{array}{cc} {[(-1+i \sqrt{3}) /(2 i)]} & {[(-1-i \sqrt{3}) /(2 i)]} \\ {[(1+i \sqrt{3}) /(2 i)]} & {[(1-i \sqrt{3}) /(2 i)]} \end{array}\right| $$ \(\mathrm{i}=\sqrt{(-1)}\) and \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}+2\) then \(\mathrm{f}(\mathrm{A})=\) (a) $$ \left|\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right| $$ (b) \([(3-\mathrm{i} \sqrt{3}) / 2]\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) (c) \([(5-\mathrm{t} \sqrt{3}) / 2]\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) (d) \(\quad(2+\mathrm{i} \sqrt{3})\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\)

Short Answer

Expert verified
The correct answer to this question is not among the given options, as the result of applying the function f(x) to the determinant of matrix A is the scalar value 3, and all the options are matrices.

Step by step solution

01

Simplify matrix A

First, simplify the matrix A using the fact that i = √(-1): $$ A=\left|\begin{array}{cc} {[-\frac{1+i \sqrt{3}}{2i}]} & {[-\frac{1-i \sqrt{3}}{2i}]} \\ {[\frac{1+i \sqrt{3}}{2i}]} & {[\frac{1-i \sqrt{3}}{2i}]} \end{array}\right| $$
02

Calculate the determinant of matrix A

We will now calculate the determinant of the simplified matrix A using the formula: det(A) = ad - bc, where a, b, c, and d are the elements of the matrix: $$ \det(A)=\left(-\frac{1+i \sqrt{3}}{2i}\right)\left(\frac{1-i \sqrt{3}}{2i}\right)-\left(-\frac{1-i \sqrt{3}}{2i}\right)\left(\frac{1+i \sqrt{3}}{2i}\right) $$ Simplify the expression by multiplying the terms: $$ \det(A)=\frac{(1+i \sqrt{3})(1-i \sqrt{3})}{4i^2}+\frac{(1-i \sqrt{3})(1+i \sqrt{3})}{4i^2} $$ Now, expand and simplify the terms further: $$ \det(A)=\frac{[(1 + i\sqrt{3} - i\sqrt{3} -3) + (1 - i\sqrt{3} + i\sqrt{3} -3)]}{4i^2} $$ $$ \det(A)=\frac{-4}{4(-1)}=1 $$ So, the determinant of matrix A is 1.
03

Apply function f(x) to the determinant

Now apply the function f(x) = x^2 + 2 to the determinant: $$ f(\det(A))=f(1)=1^2+2=3 $$
04

Compare the result with the given options

The result from Step 3 is 3. We will now match this with the four given options: (a) $$ \left|\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right| $$ This option is the identity matrix, which is not equal to the scalar value 3. (b) \([(3-i \sqrt{3}) / 2]\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) This option is also a matrix but we found a scalar value. (c) \([(5-t \sqrt{3}) / 2]\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) This option is also a matrix but we found a scalar value. (d) \((2+i \sqrt{3})\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) This option is also a matrix but we found a scalar value. None of the options match the answer.

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Most popular questions from this chapter

If \([\mathrm{x}]\) is the greatest integer less than or equal to \(\mathrm{x}\), then the determinant's value of the matrix. $$ \left|\begin{array}{ccc} {[\mathrm{e}]} & {[\pi]} & {\left[\pi^{2}-6\right]} \\ {[\pi]} & {\left[\pi^{2}-6\right]} & {[\mathrm{e}]} \\ {\left[\pi^{2}-6\right]} & {[\mathrm{e}]} & {[\pi]} \end{array}\right| \text { is } \ldots $$ (a) 8 (b) 0 (c) 1 \(-8\)

If $$ A=\left|\begin{array}{ccc} 3 a & b & c \\ b & 3 c & a \\ c & a & 3 b \end{array}\right| $$ \(\mathrm{a}, \mathrm{b}, \mathrm{c} \notin \mathrm{R}, \mathrm{abc}=1\) and \(\mathrm{AA}^{\mathrm{T}}=641\) and \(|\mathrm{A}|>0\), then \(\left(a^{3}+b^{3}+c^{3}\right)=\) (a) 343 (b) 729 (c) 256 (d) 512

If \(\mathrm{A}=\left|\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right|\) then \(\mathrm{A}^{\mathrm{n}}+(\mathrm{n}-1) \mathrm{I}=\ldots \ldots\) (a) \(2^{\mathrm{n}-1} \mathrm{~A}\) (b) \(-\mathrm{n} \mathrm{A}\) (c) \(\mathrm{nA}\) (d) \((\mathrm{n}+1) \mathrm{A}\)

If \(\mathrm{A}=\left|\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & \mathrm{a} & 1\end{array}\right|, \mathrm{A}^{-1}=\left|\begin{array}{ccc}(1 / 2) & -(1 / 2) & (1 / 2) \\ -4 & 3 & \mathrm{c} \\ (5 / 2) & -(3 / 2) & (1 / 2)\end{array}\right|\) then (a) \(\mathrm{a}=2, \mathrm{c}=-(1 / 2)\) (b) \(\mathrm{a}=1, \mathrm{c}=-1\) (c) \(\mathrm{a}=-1, \mathrm{c}=1\) (d) \(\mathrm{a}=(1 / 2), \mathrm{c}=(1 / 2)\)

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are positive and not all equal, then the value of determinant $$ \left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| \text { is ... } $$ (a) \(>0\) (b) \(\geq 0\) (c) \(<0\) \((\mathrm{d}) \leq 0\)
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