Evaluate: \\[ \\left|\\begin{array}{rrr} 2 & -1 & | Answer.AI

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Evaluate:

| \begin{array}{rrr} 2 & - 1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{array} |

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Step 1 - We can use the Laplace expansion method to evaluate the determinant of the given matrix. Let’s expand along the first row.

| \begin{array}{rrr} 2 & - 1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{array} | = 2 | \begin{array}{rr} 3 & 4 \\ 2 & 1 \end{array} | - (- 1) | \begin{array}{rr} 1 & 4 \\ 2 & 1 \end{array} | + 2 | \begin{array}{rr} 1 & 3 \\ 2 & 1 \end{array} |

Step 2 - Now, we need to evaluate the determinants of the 2x2 matrices.

| \begin{array}{rr} 3 & 4 \\ 2 & 1 \end{array} | = (3 \times 1) - (4 \times 2) = - 5

| \begin{array}{rr} 1 & 4 \\ 2 & 1 \end{array} | = (1 \times 1) - (4 \times 2) = - 7

| \begin{array}{rr} 1 & 3 \\ 2 & 1 \end{array} | = (1 \times 1) - (3 \times 2) = - 5

Substituting these values in the Laplace expansion formula, we get:

| \begin{array}{rrr} 2 & - 1 & 2 \\ 1 & 3 & 4 \\ 1 & 2 & 1 \end{array} | = 2 (- 5) - (- 1) (- 7) + 2 (- 5) = - 9

Answer - The determinant of the given matrix is -9.
Laplace expansion method for determinant.

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