The problem asks us to find the value of \( \alpha \) given a 2x2 matrix \( A \) and the condition that the determinant of \( A^3 \) is 125. We are given:
First, let's find the determinant of the matrix \( A \). For a 2x2 matrix \( \left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right] \), the determinant is given by \( ad - bc \).
For our matrix \( A \):
\( \det(A) = (\alpha)(\alpha) - (2)(2) \)
\( \det(A) = \alpha^2 - 4 \)
Using the Property of Determinants: \( \det(A^n) = (\det(A))^n \)
A useful property of determinants is that the determinant of a matrix raised to a power is equal to the determinant of the matrix raised to that same power. In this case, we have \( A^3 \), so:
\( \det(A^3) = (\det(A))^3 \)
Setting up the Equation to Find Alpha
We are given that \( \det(A^3) = 125 \). Using the property above, we can write:
\( (\det(A))^3 = 125 \)
Now, substitute the expression for \( \det(A) \) that we found:
\( (\alpha^2 - 4)^3 = 125 \)
Solving for Alpha
To solve for \( \alpha \), we first need to find the value of \( \alpha^2 - 4 \). We can do this by taking the cube root of both sides of the equation:
\( \sqrt[3]{(\alpha^2 - 4)^3} = \sqrt[3]{125} \)
\( \alpha^2 - 4 = 5 \)
Now, isolate \( \alpha^2 \):
\( \alpha^2 = 5 + 4 \)
\( \alpha^2 = 9 \)
Finally, take the square root of both sides to find the values of \( \alpha \):
\( \alpha = \pm \sqrt{9} \)
\( \alpha = \pm 3 \)
Thus, the possible values for \( \alpha \) are \( +3 \) and \( -3 \).
Step
Calculation
Result
1
Calculate \( \det(A) \)
\( \det(A) = \alpha^2 - 4 \)
2
Use \( \det(A^3) = (\det(A))^3 \)
\( (\alpha^2 - 4)^3 = 125 \)
3
Take cube root
\( \alpha^2 - 4 = 5 \)
4
Solve for \( \alpha^2 \)
\( \alpha^2 = 9 \)
5
Solve for \( \alpha \)
\( \alpha = \pm 3 \)
The value of \( \alpha \) is \( \pm 3 \).
Revision Table: Matrix Determinant Concepts
Concept
Description
Formula/Example
Determinant of 2x2 Matrix
A scalar value calculated from the elements of a square matrix.
\( \det \left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right] = ad - bc \)
Determinant of Matrix Power
The determinant of a matrix raised to a power \( n \) is the determinant raised to \( n \).
\( \det(A^n) = (\det(A))^n \)
Cube Root
The number that, when multiplied by itself three times, gives the original number.
\( \sqrt[3]{x^3} = x \). \( \sqrt[3]{125} = 5 \) because \( 5 \times 5 \times 5 = 125 \).
Additional Information: Matrix Properties and Determinants
Determinants are fundamental in linear algebra and have many important properties and applications. Here are a few points related to matrix determinants:
Singular Matrices: A matrix is called singular if its determinant is zero. Singular matrices do not have an inverse.
Invertibility: A square matrix \( A \) is invertible if and only if \( \det(A) \neq 0 \).
Geometric Interpretation: For a 2x2 matrix, the absolute value of the determinant represents the area of the parallelogram formed by the column vectors (or row vectors) of the matrix. For a 3x3 matrix, it represents the volume of the parallelepiped.
Determinant of a Product: For two square matrices \( A \) and \( B \) of the same size, \( \det(AB) = \det(A) \det(B) \). This property is related to \( \det(A^n) = (\det(A))^n \) because \( A^3 = A \cdot A \cdot A \).
Determinant of Transpose: The determinant of the transpose of a matrix is equal to the determinant of the original matrix, i.e., \( \det(A^T) = \det(A) \).
Understanding these properties helps in solving various problems involving matrices and their determinants.
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