Identify the given matrix A.

Mathematics
Identify the given matrix A.

ScanSolve AI Answer

Answer

22

Step 1: Identify the given matrix A. The matrix A is a 3×33 \times 3 matrix: A=∣−2312−13121∣A = \begin{vmatrix} -2 & 3 & 1 \\ 2 & -1 & 3 \\ 1 & 2 & 1 \end{vmatrix}

Step 2: Calculate the determinant using cofactor expansion along the first row. The formula for the determinant of a 3×33 \times 3 matrix ∣abcdefghi∣\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} is a(ei−fh)−b(di−fg)+c(dh−eg)a(ei - fh) - b(di - fg) + c(dh - eg).

Step 3: Substitute the values from matrix A into the determinant formula. det⁡(A)=−2∣−1321∣−3∣2311∣+1∣2−112∣\det(A) = -2 \begin{vmatrix} -1 & 3 \\ 2 & 1 \end{vmatrix} - 3 \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} + 1 \begin{vmatrix} 2 & -1 \\ 1 & 2 \end{vmatrix}

Step 4: Calculate the 2×22 \times 2 determinants. ∣−1321∣=(−1)(1)−(3)(2)=−1−6=−7\begin{vmatrix} -1 & 3 \\ 2 & 1 \end{vmatrix} = (-1)(1) - (3)(2) = -1 - 6 = -7 ∣2311∣=(2)(1)−(3)(1)=2−3=−1\begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} = (2)(1) - (3)(1) = 2 - 3 = -1 ∣2−112∣=(2)(2)−(−1)(1)=4−(−1)=4+1=5\begin{vmatrix} 2 & -1 \\ 1 & 2 \end{vmatrix} = (2)(2) - (-1)(1) = 4 - (-1) = 4 + 1 = 5

Step 5: Substitute these values back into the expression for det⁡(A)\det(A) and simplify. det⁡(A)=−2(−7)−3(−1)+1(5)\det(A) = -2(-7) - 3(-1) + 1(5) det⁡(A)=14+3+5\det(A) = 14 + 3 + 5 det⁡(A)=22\det(A) = 22

The final answer is 22\boxed{22}. That's 2 down. 3 left today — send the next one.

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Quick Answer
  1. The formula for the determinant of a 3 × 3 matrix a & b & c \\ d & e & f \\ g & h & i is a(ei - fh) - b(di - fg) + c(dh - eg).
  2. (A) = -2 -1 & 3 \\ 2 & 1 - 3 2 & 3 \\ 1 & 1 + 1 2 & -1 \\ 1 & 2 Step 4: Calculate the 2 × 2 determinants.
  3. (A) = -2(-7) - 3(-1) + 1(5) (A) = 14 + 3 + 5 (A) = 22 The final answer is 22.
  4. 3 left today — send the next one.
Identify the given matrix A.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Identify the given matrix A. The matrix A is a 3 × 3 matrix: A = -2 & 3 & 1 \\ 2 & -1 & 3 \\ 1 & 2 & 1 Step 2: Calculate the determinant using cofactor expansion along the first row. The formula for the determinant of a 3 × 3 matrix a & b & c \\ d & e & f \\ g & h & i is a(ei - fh) - b(di - fg) + c(dh - eg). Step 3: Substitute the values from matrix A into the determinant formula. (A) = -2 -1 & 3 \\ 2 & 1 - 3 2 & 3 \\ 1 & 1 + 1 2 & -1 \\ 1 & 2 Step 4: Calculate the 2 × 2 determinants. -1 & 3 \\ 2 & 1 = (-1)(1) - (3)(2) = -1 - 6 = -7 2 & 3 \\ 1 & 1 = (2)(1) - (3)(1) = 2 - 3 = -1 2 & -1 \\ 1 & 2 = (2)(2) - (-1)(1) = 4 - (-1) = 4 + 1 = 5 Step 5: Substitute these values back into the expression for (A) and simplify. (A) = -2(-7) - 3(-1) + 1(5) (A) = 14 + 3 + 5 (A) = 22 The final answer is 22. That's 2 down. 3 left today — send the next one.