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Zorluk: OrtaMatrices and Determinants

Given the matrices A=(2143)A = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} and B=(1−125)B = \begin{pmatrix} 1 & -1 \\ 2 & 5 \end{pmatrix}, find the determinant of the matrix C=2A−BC = 2A - B.

Cevap: -15

Cevap

The determinant of the matrix C=2A−BC = 2A - B is −15-15.
Scalar multiplication gives 2A=(4286)2A = \begin{pmatrix} 4 & 2 \\ 8 & 6 \end{pmatrix}. Subtracting BB yields C=(3361)C = \begin{pmatrix} 3 & 3 \\ 6 & 1 \end{pmatrix}. Evaluating the determinant gives det⁡(C)=(3)(1)−(3)(6)=3−18=−15\det(C) = (3)(1) - (3)(6) = 3 - 18 = -15.

Adım Adım Çözüm

1
Multiply matrix AA by scalar 22
2A=(4286)2A = \begin{pmatrix} 4 & 2 \\ 8 & 6 \end{pmatrix}
Scalar multiplication requires multiplying each entry of matrix AA by 22.
2
Subtract matrix BB from matrix 2A2A entry-wise to find matrix CC
C=(4−12−(−1)8−26−5)=(3361)C = \begin{pmatrix} 4 - 1 & 2 - (-1) \\ 8 - 2 & 6 - 5 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 6 & 1 \end{pmatrix}
Subtract corresponding entries of matrix BB from 2A2A.
3
Calculate the determinant of matrix CC
det⁡(C)=(3)(1)−(3)(6)=3−18=−15\det(C) = (3)(1) - (3)(6) = 3 - 18 = -15
The determinant of a 2×22 \times 2 matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} is ad−bcad - bc.

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Matrix Operations and Determinants
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Matrices and Determinants

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