Determinant of a 3x3 Matrix - Examples with Answers

To find the determinant of the matrix $latex B$, we can use the cofactor expansion method along the first row.

Then, if we denote the determinant by $latex |B|$ or $latex \det(B)$.

$latex |B| = 2 \cdot C_{11} - 5 \cdot C_{12} + 3 \cdot C_{13} $

Where $latex C_{11}$, $latex C_{12}$, and $latex C_{13}$ are the cofactors of the elements of the first row.

To find the cofactors, we need to calculate the determinants of the 2x2 matrices obtained by eliminating the row and column of each element of the first row:

$$ C_{11} = \begin{vmatrix} 4 & 6 \\ 8 & 9 \end{vmatrix} $$

$$= (4 \cdot 9) - (6 \cdot 8)$$

$$= 36 - 48 = -12$$

$$ C_{12} = \begin{vmatrix} 1 & 6 \\ 7 & 9 \end{vmatrix}$$

$$ = (1 \cdot 9) - (6 \cdot 7) $$

$$= 9 - 42 = -33 $$

$$ C_{13} = \begin{vmatrix} 1 & 4 \\ 7 & 8 \end{vmatrix} $$

$$= (1 \cdot 8) - (4 \cdot 7) $$

$$= 8 - 28 = -20$$

Now, we substitute the values of the cofactors into the determinant formula:

$$ |B| = 2 \cdot (-12) - 5 \cdot (-33) + 3 \cdot (-20) $$

$$= -24 + 165 - 60 = 81 $$

Therefore, the determinant of the 3x3 matrix is 81.

We can take the first row and form the following formula. Remember that we multiply each element by the determinant of the corresponding minor.

$$ |B| = b_{11}(b_{22}b_{33} - b_{23}b_{32}) - b_{12}(b_{21}b_{33} - b_{23}b_{31}) + b_{13}(b_{21}b_{32} - b_{22}b_{31}) $$

Substituting the values of matrix B, we have:

$$ |B| = 1(5 \cdot 9 - 8 \cdot 6) - 4(2 \cdot 9 - 8 \cdot 3) + 7(2 \cdot 6 - 5 \cdot 3) $$

Simplifying, we have:

$$ |B| = 1(-3) - 4(-6) + 7(-3) $$

$$ |B| = -3 + 24 - 21 $$

$latex |B| = 0 $

Similar to the previous examples, we will obtain a formula using the cofactor expansion method in the first row:

$$|C| = c_{11}(c_{22}c_{33} - c_{23}c_{32}) - c_{12}(c_{21}c_{33} - c_{23}c_{31}) + c_{13}(c_{21}c_{32} - c_{22}c_{31})$$

Using the values of the elements of the matrix C, we have:

$$|C| = 3(1 \cdot 9 - 5 \cdot 8) - 0(4 \cdot 9 - 5 \cdot 7) + 2(4 \cdot 8 - 1 \cdot 7)$$

Performing the operations, we have:

$latex |C| = 3(-31) - 0(-1) + 2(25)$

$latex |C| = -93 + 0 + 50$

$latex |C| = -43$

Using the first row of the matrix, we form the following formula:

$$|D| = d_{11}(d_{22}d_{33} - d_{23}d_{32}) - d_{12}(d_{21}d_{33} - d_{23}d_{31}) + d_{13}(d_{21}d_{32} - d_{22}d_{31})$$

Substituting the values of the elements of the matrix D in this formula, we have:

$$|D| = -1(-5 \cdot -9 - 6 \cdot 8) - 2(4 \cdot -9 - 6 \cdot -7) + (-3)(4 \cdot 8 - (-5) \cdot -7)$$

Solving the operations and simplifying, we have:

$$|D| = -1(45 - 48) - 2(-36 + 42) + (-3)(32 + 35)$$

$$|D| = -1(-3) - 2(-6) + (-3)(67)$$

$latex |D| = 3 + 12 - 201$

$latex |D| = -186$

Similar to the previous exercises, we will select row 1 to apply the cofactor method:

$$|E| = e_{11}(e_{22}e_{33} - e_{23}e_{32}) - e_{12}(e_{21}e_{33} - e_{23}e_{31}) + e_{13}(e_{21}e_{32} - e_{22}e_{31})$$

Using the values of the elements of the matrix E in the formula, we have:

$$|E| = 2(3 \cdot -6 - (-7) \cdot 5) - (-4)(-1 \cdot -6 - (-7) \cdot 4) + 6(-1 \cdot 5 - 3 \cdot 4)$$

Solving the operations, we have:

$$|E| = 2(-18 + 35) - (-4)(6 + 28) + 6(-5 - 12)$$

$$|E| = 2(17) - (-4)(34) + 6(-17)$$

$$|E| = 34 + 136 - 102$$

$latex |E| = 68$

Using the cofactor expansion method, we can form the following formula using the first row of the matrix:

$$|F| = f_{11}(f_{22}f_{33} - f_{23}f_{32}) - f_{12}(f_{21}f_{33} - f_{23}f_{31}) + f_{13}(f_{21}f_{32} - f_{22}f_{31})$$

Using the values of the elements of the matrix F, we have:

$$|F| = 3(4 \cdot -3 - (-5) \cdot 6) - (-2)(-1 \cdot -3 - (-5) \cdot 2) + 1(-1 \cdot 6 - 4 \cdot 2)$$

Solving the operations, we have:

$$|F| = 3(-12 + 30) - (-2)(3 + 10) + 1(-6 - 8)$$

$$|F| = 3(18) - (-2)(13) + 1(-14)$$

$latex |F| = 54 + 26 - 14$

$latex |F| = 66$

If we take the first row of the matrix, we can form the following formula:

$$|G| = g_{11}(g_{22}g_{33} - g_{23}g_{32}) - g_{12}(g_{21}g_{33} - g_{23}g_{31}) + g_{13}(g_{21}g_{32} - g_{22}g_{31})$$

Using this formula with the values of the elements of the matrix G, we have:

$$|G| = -4(-2 \cdot 8 - 6 \cdot 1) - 7(5 \cdot 8 - 6 \cdot -3) + (-1)(5 \cdot 1 - (-2) \cdot -3)$$

Simplifying this expression, we have:

$$|G| = -4(-16 - 6) - 7(40 + 18) + (-1)(5 - 6)$$

$$|G| = -4(-22) - 7(58) + (-1)(-1)$$

$latex |G| = 88 - 406 + 1$

$latex |G| = -317$

By taking the first row of the matrix, we can multiply each element by the determinant of the smallest to form the following formula:

$$|H| = h_{11}(h_{22}h_{33} - h_{23}h_{32}) - h_{12}(h_{21}h_{33} - h_{23}h_{31}) + h_{13}(h_{21}h_{32} - h_{22}h_{31})$$

Substituting the values of the elements of the matrix H, we have:

$$|H| = 1(1 \cdot 1 - (-4) \cdot (-1)) - 2(-2 \cdot 1 - (-4) \cdot 3) + 3(-2 \cdot (-1) - 1 \cdot 3)$$

Now, we solve the operations and simplify:

$$|H| = 1(1 - 4) - 2(-2 + 12) + 3(2 - 3)$$

$latex |H| = 1(-3) - 2(10) + 3(-1)$

$latex |H| = -3 - 20 - 3$

$latex |H| = -26$

We form the following formula by taking the first row of the I matrix:

$$|I| = i_{11}(i_{22}i_{33} - i_{23}i_{32}) - i_{12}(i_{21}i_{33} - i_{23}i_{31}) + i_{13}(i_{21}i_{32} - i_{22}i_{31})$$

Substituting the values of matrix I, we have:

$$|I| = 2(4 \cdot 1 - 2 \cdot -2) - 3(-1 \cdot 1 - 2 \cdot 3) + (-1)(-1 \cdot -2 - 4 \cdot 3)$$

Solving operations and simplifying:

$$|I| = 2(4 + 4) - 3(-1 - 6) + (-1)(2 - 12)$$

$latex |I| = 2(8) - 3(-7) + (-1)(-10)$

$latex |I| = 16 + 21 + 10$

$latex |I| = 47$

Using the first row of the matrix, we form the following formula:

$$|J| = j_{11}(j_{22}j_{33} - j_{23}j_{32}) - j_{12}(j_{21}j_{33} - j_{23}j_{31}) + j_{13}(j_{21}j_{32} - j_{22}j_{31})$$

Now, we use the values of the matrix elements:

$$|J| = -1(2 \cdot -2 - (-1) \cdot 1) - 3(3 \cdot -2 - (-1) \cdot 1) + 5(3 \cdot 1 - 2 \cdot 1)$$

Solving the operations, we have

$$|J| = -1(-4 - 1) - 3(-6 + 1) + 5(3 - 2)$$

$latex |J| = -1(-5) - 3(-5) + 5(1)$

$latex |J| = 5 + 15 + 5$

$latex |J| = 25$