Math Problem Statement

Solve this

If A and B are given matrices, calculate the determinant of BA (|BA|). Further, if it is a non-singular matrix, find the inverse of BA.

Solution

To solve this problem, we need to compute the following:

  1. Multiply the two matrices AA and BB to get BABA.
  2. Compute the determinant of BABA, denoted as ∣BA∣|BA|.
  3. Check if BABA is non-singular (i.e., ∣BA∣≠0|BA| \neq 0).
  4. If BABA is non-singular, compute its inverse.

Let's perform the operations step-by-step.

1. Matrices AA and BB:

Given:

\begin{bmatrix} 2 & 1 & 2 & 1 & 2 \\ 3 & 1 & 2 & 1 & 3 \\ 1 & 2 & 1 & 4 & 2 \\ 2 & 1 & 1 & 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 & 3 & 1 \\ 1 & 1 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 2 & 0 \\ 2 & 0 & 1 & 1 \end{bmatrix}.$$ We aim to compute: $$BA = B \times A$$ --- ### 2. Matrix Multiplication $$BA$$: The dimensions of $$B$$ are $$4 \times 5$$, and the dimensions of $$A$$ are $$5 \times 4$$. Therefore, the result $$BA$$ will be a $$4 \times 4$$ matrix. Let me calculate $$BA$$. numpy as np # Define matrices A and B A = np.array([[2, 1, 2, 1, 2], [3, 1, 2, 1, 3], [1, 2, 1, 4, 2], [2, 1, 1, 2, 1]]) B = np.array([[1, 0, 3, 1], [1, 1, 0, 2], [0, 1, 0, 1], [1, 0, 2, 0], [2, 0, 1, 1]]) # Calculate BA BA = np.dot(B, A) BAThe resulting matrix $$BA$$ is: $$BA = \begin{bmatrix} 7 & 8 & 6 & 15 & 9 \\ 9 & 4 & 6 & 6 & 7 \\ 5 & 2 & 3 & 3 & 4 \\ 4 & 5 & 4 & 9 & 6 \\ 7 & 5 & 6 & 8 & 7 \end{bmatrix}.$$

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Determinants
Inverse of a Matrix

Formulas

Matrix multiplication formula: C = AB where C[i][j] = sum(A[i][k] * B[k][j])
Determinant formula for a 2x2 matrix: det(A) = ad - bc
Inverse of a matrix: A⁻¹ exists if |A| ≠ 0 and is computed as adj(A)/|A|

Theorems

Properties of Determinants
Invertibility of Matrices

Suitable Grade Level

Undergraduate (or Advanced High School for Linear Algebra)

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