Calculating the Determinant of a Matrix with Powers of 'i'

The problem asks us to find the determinant of a 3x3 matrix where the elements are different powers of the imaginary unit \(\rm i\), where \(\rm i = \sqrt {-1}\).

Understanding Powers of the Imaginary Unit 'i'

The powers of \(\rm i\) follow a repeating cycle of four:

• \(\rm i^1 = i\)
• \(\rm i^2 = -1\)
• \(\rm i^3 = i^2 \cdot i = -1 \cdot i = -i\)
• \(\rm i^4 = (i^2)^2 = (-1)^2 = 1\)
• \(\rm i^5 = i^4 \cdot i = 1 \cdot i = i\)

In general, for any integer exponent \(n\), the value of \(\rm i^n\) can be found by looking at the remainder when \(n\) is divided by 4:

• If \(n \pmod 4 = 1\), then \(\rm i^n = i\)
• If \(n \pmod 4 = 2\), then \(\rm i^n = -1\)
• If \(n \pmod 4 = 3\), then \(\rm i^n = -i\)
• If \(n \pmod 4 = 0\), then \(\rm i^n = 1\)

Simplifying the Matrix Elements

Let's simplify each element of the given matrix \(\left| {\begin{array}{*{20}{c}} {\rm{i}}&{{{\rm{i}}^2}}&{{{\rm{i}}^3}}\\ {{{\rm{i}}^4}}&{{{\rm{i}}^6}}&{{{\rm{i}}^8}}\\ {{{\rm{i}}^9}}&{{{\rm{i}}^{12}}}&{{{\rm{i}}^{15}}} \end{array}} \right|\) using the properties of powers of \(\rm i\):

• \(\rm i^1 = i\)
• \(\rm i^2 = -1\)
• \(\rm i^3 = -i\)
• \(\rm i^4 = 1\)
• \(\rm i^6 = i^{4+2} = i^2 = -1\)
• \(\rm i^8 = i^{2 \times 4} = (i^4)^2 = 1^2 = 1\)
• \(\rm i^9 = i^{8+1} = i^1 = i\)
• \(\rm i^{12} = i^{3 \times 4} = (i^4)^3 = 1^3 = 1\)
• \(\rm i^{15} = i^{12+3} = i^3 = -i\)

Substituting these values into the matrix, we get the simplified matrix:

So the determinant we need to calculate is \(\left| {\begin{array}{*{20}{c}} {\rm{i}}&{-1}&{-\rm{i}}\\ {1}&{-1}&{1}\\ {\rm{i}}&{1}&{-\rm{i}}} \end{array}} \right|\).

Calculating the Determinant

We can calculate the determinant of this 3x3 matrix using the cofactor expansion method. Let's expand along the first row:

Determinant \(= \rm i \cdot C_{11} + (-1) \cdot C_{12} + (-i) \cdot C_{13}\)

Where \(C_{ij} = (-1)^{i+j} M_{ij}\) is the cofactor and \(M_{ij}\) is the minor determinant obtained by removing the \(i\)-th row and \(j\)-th column.

Let's calculate the minors and cofactors:

1. For element \(\rm i\) (Row 1, Column 1):
   Minor \(M_{11} = \left| {\begin{array}{*{20}{c}} -1&{1}\\ {1}&{-\rm{i}} \end{array}} \right| = (-1)(-\rm i) - (1)(1) = \rm i - 1\)
   Cofactor \(C_{11} = (-1)^{1+1} M_{11} = (1)(\rm i - 1) = i - 1\)
2. For element \(-1\) (Row 1, Column 2):
   Minor \(M_{12} = \left| {\begin{array}{*{20}{c}} 1&{1}\\ {\rm{i}}&{-\rm{i}} \end{array}} \right| = (1)(-\rm i) - (1)(\rm i) = -\rm i - i = -2i\)
   Cofactor \(C_{12} = (-1)^{1+2} M_{12} = (-1)(-2\rm i) = 2i\)
3. For element \(\rm -i\) (Row 1, Column 3):
   Minor \(M_{13} = \left| {\begin{array}{*{20}{c}} 1&{-1}\\ {\rm{i}}&{1} \end{array}} \right| = (1)(1) - (-1)(\rm i) = 1 + i\)
   Cofactor \(C_{13} = (-1)^{1+3} M_{13} = (1)(1 + \rm i) = 1 + i\)

Now, substitute the cofactors back into the determinant formula:

Determinant \(= \rm i \cdot C_{11} + (-1) \cdot C_{12} + (-i) \cdot C_{13}\)

Determinant \(= \rm i \cdot (\rm i - 1) + (-1) \cdot (2i) + (-i) \cdot (1 + i)\)

Expand the terms:

Determinant \(= (\rm i \cdot \rm i) - (\rm i \cdot 1) - 2i - (\rm i \cdot 1) - (\rm i \cdot i)\)

Determinant \(= \rm i^2 - i - 2i - i - i^2\)

Substitute \(\rm i^2 = -1\):

Determinant \(= (-1) - \rm i - 2i - i - (-1)\)

Determinant \(= -1 - \rm i - 2i - i + 1\)

Combine the real and imaginary parts:

Determinant \(= (-1 + 1) + (-\rm i - 2i - i)\)

Determinant \(= 0 + (-4\rm i)\)

Determinant \(= -4\rm i\)

The value of the determinant is \(-4\rm i\).

Revision Table: Powers of i

Additional Information: Determinant Properties

Determinants are fundamental in linear algebra and have several useful properties:

• The determinant of an identity matrix is 1.
• The determinant of a zero matrix is 0.
• If a matrix has a row or a column of zeros, its determinant is 0.
• If a matrix has two identical rows or columns, its determinant is 0.
• If one row (or column) is a scalar multiple of another row (or column), the determinant is 0.
• Swapping two rows or columns changes the sign of the determinant.
• Multiplying a single row or column by a scalar \(k\) multiplies the determinant by \(k\).
• Adding a multiple of one row to another row (or column to column) does not change the determinant.
• The determinant of the product of two matrices is the product of their determinants: \(\det(AB) = \det(A) \det(B)\).
• The determinant of the transpose of a matrix is equal to the determinant of the original matrix: \(\det(A^T) = \det(A)\).
• A square matrix is invertible if and only if its determinant is non-zero.

In this problem, while the matrix elements are complex numbers, the method for calculating the determinant remains the same as for real number matrices.