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Difficulty: MediumMatrices and Determinants

Given the matrices A=(2134)A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} and B=(1−203)B = \begin{pmatrix} 1 & -2 \\ 0 & 3 \end{pmatrix}, if C=AB−2IC = AB - 2I, where II is the 2×22 \times 2 identity matrix, what is the determinant of matrix CC?

  1. 33Answer
  2. B
    −3-3
  3. C
    00
  4. D
    1515

Answer

The determinant of matrix C is 3.
Multiplying matrices A and B using standard row-by-column multiplication yields \begin{pmatrix} 2 & -1 \\ 3 & 6 \end{pmatrix}. Subtracting the scaled identity matrix \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} produces matrix C = \begin{pmatrix} 0 & -1 \\ 3 & 4 \end{pmatrix}. Taking the determinant ad - bc gives (0)(4) - (-1)(3) = 3.

Step-by-Step Solution

1
Compute the matrix product AB.
AB = \begin{pmatrix} (2)(1) + (1)(0) & (2)(-2) + (1)(3) \\ (3)(1) + (4)(0) & (3)(-2) + (4)(3) \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ 3 & 6 \end{pmatrix}
Matrix multiplication uses dot products of rows of the first matrix and columns of the second matrix.
2
Subtract 2I from AB to find matrix C.
C = \begin{pmatrix} 2 & -1 \\ 3 & 6 \end{pmatrix} - \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 3 & 4 \end{pmatrix}
The identity matrix I multiplied by scalar 2 has 2 on its main diagonal and 0 elsewhere.
3
Calculate the determinant of matrix C.
\det(C) = (0)(4) - (-1)(3) = 0 + 3 = 3
For a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix}, the determinant is ad - bc.

Key Concept

Matrix multiplication, matrix arithmetic operations, and evaluation of 2x2 determinants.
Estimated Time:1m 30s
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Topics

Matrices and Determinants

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