Math Problem Statement

Evaluate the determinant of the 3x3 matrix: | j^2 2 j | | 1+j 5 1 | | -3 -j4 0 |

Solution

The image you uploaded seems to be a mathematical problem involving complex numbers. Specifically, it appears to be asking to evaluate a determinant or a matrix involving complex numbers (where jj is the imaginary unit, often written as ii in some contexts).

Let's break it down step by step:

We are tasked to evaluate the determinant of a 3x3 matrix involving complex numbers. The matrix looks like this:

j^2 & 2 & j \\ (1+j) & 5 & 1 \\ -3 & -j4 & 0 \\ \end{array} \right|$$ To solve this, we will need to calculate the determinant using the cofactor expansion or the standard determinant formula for a 3x3 matrix. Let me proceed with calculating the determinant for you. sympy as sp # Define the matrix with the elements given in the problem j = sp.I # Imaginary unit matrix = sp.Matrix([ [j**2, 2, j], [1+j, 5, 1], [-3, -j*4, 0] ]) # Calculate the determinant determinant = matrix.det() determinant.simplify()The determinant of the given matrix is: $$-2 + 15j$$ Would you like a detailed explanation of the steps or any further questions? Here are some related topics you may want to explore: 1. How to calculate the determinant of a 3x3 matrix. 2. The properties of determinants involving complex numbers. 3. An explanation of the imaginary unit $$j$$ in complex number operations. 4. How to perform matrix operations involving complex numbers. 5. The concept of cofactor expansion in calculating determinants. **Tip**: When dealing with matrices that include complex numbers, it’s helpful to remember that $$j^2 = -1$$, which can simplify expressions involving powers of $$j$$.

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Math Problem Analysis

Mathematical Concepts

Matrix Determinants
Complex Numbers
Linear Algebra

Formulas

Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Properties of complex numbers: j^2 = -1

Theorems

Determinant Expansion by Cofactors
Basic Properties of Determinants

Suitable Grade Level

Undergraduate Level (Linear Algebra and Complex Numbers)