Examkin

Question

Difficulty: MediumMatrices and Determinants

Given the matrices M=(2−134)M = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} and N=(1023)N = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}, what is the determinant of the matrix product MNMN?

  1. A
    14
  2. B
    15
  3. C
    24
  4. 33Answer

Answer

33
Using the property that the determinant of a product of matrices equals the product of their individual determinants, det⁡(M)=(2)(4)−(−1)(3)=11\det(M) = (2)(4) - (-1)(3) = 11 and det⁡(N)=(1)(3)−(0)(2)=3\det(N) = (1)(3) - (0)(2) = 3. Multiplying these gives det⁡(MN)=11×3=33\det(MN) = 11 \times 3 = 33.

Step-by-Step Solution

1
Calculate the determinant of matrix MM
\det(M) = (2)(4) - (-1)(3) = 8 + 3 = 11
The determinant of a 2×22 \times 2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} is given by ad−bcad - bc.
2
Calculate the determinant of matrix NN
\det(N) = (1)(3) - (0)(2) = 3 - 0 = 3
Apply the determinant formula ad−bcad - bc to matrix NN.
3
Apply the determinant product property to find det⁡(MN)\det(MN)
\det(MN) = \det(M) \times \det(N) = 11 \times 3 = 33
For any two square matrices of the same order, det⁡(MN)=det⁡(M)⋅det⁡(N)\det(MN) = \det(M) \cdot \det(N).

Key Concept

Determinant Property of Matrix Products
Estimated Time:1m 30s
Rate this question

Topics

Matrices and Determinants

More questions from this topic