Solved on Jan 08, 2024

Find the determinant of the $3 \times 3$ matrix with entries $-1, 1, 0; 1, 0, 2; 1, -3, -1$ .

STEP 1

Assumptions
1. We need to calculate the determinant of a 3x3 matrix.
2. The matrix is given by: $\begin{pmatrix} -1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & -3 & -1 \end{pmatrix}$

STEP 2

The determinant of a 3x3 matrix $A$ , given by $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ is calculated using the formula: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

STEP 3

Now, we will apply this formula to the given matrix. Let's identify the elements of the matrix as follows: $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} -1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & -3 & -1 \end{pmatrix}$

STEP 4

Using the elements from the given matrix, we substitute them into the determinant formula: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

STEP 5

Substitute the values $a = -1$ , $b = 1$ , $c = 0$ , $d = 1$ , $e = 0$ , $f = 2$ , $g = 1$ , $h = -3$ , and $i = -1$ into the determinant formula.

STEP 6

Calculate the determinant using the substituted values: $|A| = (-1)(0 \cdot (-1) - 2 \cdot (-3)) - (1)(1 \cdot (-1) - 2 \cdot 1) + (0)(1 \cdot (-3) - 0 \cdot 1)$

STEP 7

Simplify the expression by performing the multiplications: $|A| = (-1)(0 + 6) - (1)(-1 - 2) + (0)(-3 - 0)$

STEP 8

Further simplify the expression: $|A| = (-1)(6) - (1)(-3) + (0)(-3)$

STEP 9

Calculate the values of each term: $|A| = -6 + 3 + 0$

STEP 10

Add the terms together to find the determinant: $|A| = -6 + 3 = -3$
The value of the determinant is $-3$ .

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Find the determinant of the $3 \times 3$ matrix with entries $-1, 1, 0; 1, 0, 2; 1, -3, -1$ .

STEP 1

Assumptions
1. We need to calculate the determinant of a 3x3 matrix.
2. The matrix is given by: $\begin{pmatrix} -1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & -3 & -1 \end{pmatrix}$

STEP 2

The determinant of a 3x3 matrix $A$ , given by $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ is calculated using the formula: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

STEP 3

Now, we will apply this formula to the given matrix. Let's identify the elements of the matrix as follows: $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} -1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & -3 & -1 \end{pmatrix}$

STEP 4

Using the elements from the given matrix, we substitute them into the determinant formula: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

STEP 5

Substitute the values $a = -1$ , $b = 1$ , $c = 0$ , $d = 1$ , $e = 0$ , $f = 2$ , $g = 1$ , $h = -3$ , and $i = -1$ into the determinant formula.

STEP 6

Calculate the determinant using the substituted values: $|A| = (-1)(0 \cdot (-1) - 2 \cdot (-3)) - (1)(1 \cdot (-1) - 2 \cdot 1) + (0)(1 \cdot (-3) - 0 \cdot 1)$

STEP 7

Simplify the expression by performing the multiplications: $|A| = (-1)(0 + 6) - (1)(-1 - 2) + (0)(-3 - 0)$

STEP 8

Further simplify the expression: $|A| = (-1)(6) - (1)(-3) + (0)(-3)$

STEP 9

Calculate the values of each term: $|A| = -6 + 3 + 0$

STEP 10

Add the terms together to find the determinant: $|A| = -6 + 3 = -3$
The value of the determinant is $-3$ .

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Find the determinant of the 3×33 \times 33×3 matrix with entries −1,1,0;1,0,2;1,−3,−1-1, 1, 0; 1, 0, 2; 1, -3, -1−1,1,0;1,0,2;1,−3,−1.

STEP 1

Assumptions1. We need to calculate the determinant of a 3x3 matrix.2. The matrix is given by: (−1101021−3−1) \begin{pmatrix} -1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & -3 & -1 \end{pmatrix} ⎝⎛​−111​10−3​02−1​⎠⎞​

STEP 2

STEP 3

STEP 4

Using the elements from the given matrix, we substitute them into the determinant formula: ∣A∣=a(ei−fh)−b(di−fg)+c(dh−eg) |A| = a(ei - fh) - b(di - fg) + c(dh - eg) ∣A∣=a(ei−fh)−b(di−fg)+c(dh−eg)

STEP 5

Substitute the values a=−1a = -1a=−1, b=1b = 1b=1, c=0c = 0c=0, d=1d = 1d=1, e=0e = 0e=0, f=2f = 2f=2, g=1g = 1g=1, h=−3h = -3h=−3, and i=−1i = -1i=−1 into the determinant formula.

STEP 6

STEP 7

Simplify the expression by performing the multiplications: ∣A∣=(−1)(0+6)−(1)(−1−2)+(0)(−3−0) |A| = (-1)(0 + 6) - (1)(-1 - 2) + (0)(-3 - 0) ∣A∣=(−1)(0+6)−(1)(−1−2)+(0)(−3−0)

STEP 8

Further simplify the expression: ∣A∣=(−1)(6)−(1)(−3)+(0)(−3) |A| = (-1)(6) - (1)(-3) + (0)(-3) ∣A∣=(−1)(6)−(1)(−3)+(0)(−3)

STEP 9

Calculate the values of each term: ∣A∣=−6+3+0 |A| = -6 + 3 + 0 ∣A∣=−6+3+0

STEP 10

Add the terms together to find the determinant: ∣A∣=−6+3=−3 |A| = -6 + 3 = -3 ∣A∣=−6+3=−3The value of the determinant is −3-3−3.

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Find the determinant of the $3 \times 3$ matrix with entries $-1, 1, 0; 1, 0, 2; 1, -3, -1$ .

Assumptions
1. We need to calculate the determinant of a 3x3 matrix.
2. The matrix is given by: $\begin{pmatrix} -1 & 1 & 0 \\ 1 & 0 & 2 \\ 1 & -3 & -1 \end{pmatrix}$

Using the elements from the given matrix, we substitute them into the determinant formula: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

Substitute the values $a = -1$ , $b = 1$ , $c = 0$ , $d = 1$ , $e = 0$ , $f = 2$ , $g = 1$ , $h = -3$ , and $i = -1$ into the determinant formula.

Simplify the expression by performing the multiplications: $|A| = (-1)(0 + 6) - (1)(-1 - 2) + (0)(-3 - 0)$

Further simplify the expression: $|A| = (-1)(6) - (1)(-3) + (0)(-3)$

Calculate the values of each term: $|A| = -6 + 3 + 0$

Add the terms together to find the determinant: $|A| = -6 + 3 = -3$
The value of the determinant is $-3$ .