Matrix Determinant: Find The Value Of B

When you're diving into the world of linear algebra, one of the most fundamental concepts you'll encounter is the determinant of a matrix. It's a scalar value that can be computed from the elements of a square matrix, and it tells us a lot about the matrix itself. For instance, a non-zero determinant indicates that the matrix is invertible, meaning it has a unique solution for systems of linear equations. Today, we're going to explore a specific problem where we're given a 3x3 matrix and its determinant, and our mission is to find the value of an unknown variable, b, within that matrix. This type of problem is excellent for solidifying your understanding of how determinants are calculated and how they relate to the matrix's components.

Let's consider the matrix given:

$$A = \begin{bmatrix} -5 & 4 & 1 \\ 5 & -7 & -4 \\ -6 & b & 6 \end{bmatrix}$$

We are told that the determinant of this matrix, denoted as $det(A)$ or $|A|$, is equal to 24. Our goal is to solve for b. There are several ways to calculate the determinant of a 3x3 matrix. One common method is using the cofactor expansion. We can expand along any row or any column. Let's choose to expand along the first row, as it often simplifies calculations.

The formula for the determinant of a 3x3 matrix using cofactor expansion along the first row is:

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

In our case, $a_{11} = -5$, $a_{12} = 4$, and $a_{13} = 1$. The other elements are $a_{21} = 5$, $a_{22} = -7$, $a_{23} = -4$, and $a_{31} = -6$, $a_{32} = b$, and $a_{33} = 6$.

Now, let's plug these values into the formula:

$det(A) = -5((-7)(6) - (-4)(b)) - 4((5)(6) - (-4)(-6)) + 1((5)(b) - (-7)(-6))$

Let's simplify each part:

First term: $-5((-7)(6) - (-4)(b)) = -5(-42 + 4b) = 210 - 20b$

Second term: $-4((5)(6) - (-4)(-6)) = -4(30 - 24) = -4(6) = -24$

Third term: $1((5)(b) - (-7)(-6)) = 1(5b - 42) = 5b - 42$

Now, we add these terms together to get the determinant:

$det(A) = (210 - 20b) + (-24) + (5b - 42)$

Combine like terms:

$det(A) = 210 - 24 - 42 - 20b + 5b$

$det(A) = 144 - 15b$

We are given that the determinant is 24. So, we set our expression for the determinant equal to 24:

$144 - 15b = 24$

Now, we need to solve for b. Let's isolate the term with b:

$-15b = 24 - 144$

$-15b = -120$

Finally, divide by -15 to find the value of b:

$b = \frac{-120}{-15}$

$b = 8$

So, the value of b is 8. This means that if we substitute 8 for b in the original matrix, its determinant will be exactly 24. It's always a good idea to double-check your calculations, especially when dealing with negative signs and multiple steps. Let's quickly verify this by plugging b=8 back into our determinant formula: $144 - 15(8) = 144 - 120 = 24$. This confirms our result!

Understanding Determinants and Their Properties

The determinant of a matrix is a fascinating mathematical object that provides crucial information about the linear transformation represented by the matrix. For a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is simply $ad - bc$. This value represents the scaling factor of the area of a unit square under the transformation. For larger matrices, like the 3x3 matrix we're working with, the concept extends. The determinant of a 3x3 matrix can be visualized as the signed volume of the parallelepiped formed by the column (or row) vectors of the matrix. A positive determinant means the orientation of the basis vectors is preserved, while a negative determinant means it's reversed. A determinant of zero implies that the matrix is singular, meaning the vectors are linearly dependent, and the transformation collapses space into a lower dimension (a line or a point).

Our problem involved finding an unknown value, b, within a 3x3 matrix, given that its determinant is 24. We used the cofactor expansion method, which is a systematic way to break down the calculation of a larger determinant into calculations of smaller determinants (2x2 in this case). The general formula for cofactor expansion along the first row is $|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$, where $C_{ij} = (-1)^{i+j}M_{ij}$ is the cofactor and $M_{ij}$ is the minor (the determinant of the submatrix obtained by deleting the i-th row and j-th column). This method, while effective, can be prone to arithmetic errors, especially with negative numbers. That's why careful calculation and verification are key.

Another method for calculating the determinant of a 3x3 matrix is the Sarrus' rule. This rule is specific to 3x3 matrices and can sometimes be quicker if you remember it. To use Sarrus' rule, you rewrite the first two columns of the matrix to the right of the matrix:

$$\begin{bmatrix} -5 & 4 & 1 \\ 5 & -7 & -4 \\ -6 & b & 6 \end{bmatrix} \begin{matrix} -5 & 4 \\ 5 & -7 \\ -6 & b \end{matrix}$$

Then, you sum the products of the diagonals going from top-left to bottom-right and subtract the sum of the products of the diagonals going from top-right to bottom-left:

$det(A) = [(-5)(-7)(6) + (4)(-4)(-6) + (1)(5)(b)] - [(1)(-7)(-6) + (-5)(-4)(b) + (4)(5)(6)]$

Let's calculate this:

First part: $(-5)(-7)(6) = 210$; $(4)(-4)(-6) = 96$; $(1)(5)(b) = 5b$. Sum = $210 + 96 + 5b = 306 + 5b$.

Second part: $(1)(-7)(-6) = 42$; $(-5)(-4)(b) = 20b$; $(4)(5)(6) = 120$. Sum = $42 + 20b + 120 = 162 + 20b$.

Now, subtract the second sum from the first:

$det(A) = (306 + 5b) - (162 + 20b)$

$det(A) = 306 + 5b - 162 - 20b$

$det(A) = 144 - 15b$

We arrive at the same expression for the determinant as with the cofactor expansion method. Setting this equal to 24:

$144 - 15b = 24$

$-15b = 24 - 144$

$-15b = -120$

$b = \frac{-120}{-15}$

$b = 8$

Both methods yield the same result, reinforcing the correctness of our answer. Understanding these different methods for determinant calculation is vital for a comprehensive grasp of matrix operations. Each method has its own advantages and potential pitfalls, and knowing when to use which can streamline your problem-solving process.

Why Determinants Matter in Mathematics

The determinant of a matrix is far more than just a number calculated from a grid of numbers; it's a concept deeply interwoven into various branches of mathematics and its applications. In linear algebra, the determinant is a key indicator of a matrix's properties. As mentioned, a non-zero determinant signifies that a matrix is invertible, which is fundamental for solving systems of linear equations using methods like Cramer's Rule. If a system of equations can be represented as $Ax = c$, where $A$ is the coefficient matrix, $x$ is the vector of variables, and $c$ is the constant vector, then if $det(A) eq 0$, a unique solution $x = A^{-1}c$ exists.

Beyond solving equations, determinants play a crucial role in understanding geometric transformations. When a matrix represents a linear transformation in Euclidean space, its determinant tells us how the volume (or area in 2D) of geometric shapes changes under that transformation. A determinant of 2, for instance, means the transformation doubles the volume of any object. A negative determinant indicates a reflection or inversion of orientation, in addition to a scaling of volume. A zero determinant implies that the transformation collapses the space into a lower dimension, meaning vectors are no longer linearly independent, and the matrix cannot be inverted.

In calculus, particularly in multivariable calculus, the determinant appears in the Jacobian determinant, which is used for changing variables in multiple integrals. The Jacobian acts as a local scaling factor for the volume element during a change of coordinates, analogous to how the determinant scales volumes for linear transformations. This is essential for correctly evaluating integrals over complex regions.

Furthermore, determinants are fundamental in the study of eigenvalues and eigenvectors. The characteristic equation of a matrix $A$, which is $det(A - \lambda I) = 0$ (where $\lambda$ represents eigenvalues and $I$ is the identity matrix), is a polynomial equation derived from the determinant. The roots of this equation are the eigenvalues of the matrix, which are critical for understanding the behavior of dynamic systems, stability analysis, and quantum mechanics.

In our specific problem, finding the value of b given the determinant was a straightforward application of determinant calculation. The process involved expanding the determinant, setting it equal to the given value (24), and then solving the resulting linear equation for b. The answer, $b=8$, means that the specific matrix with this value of b possesses properties associated with a determinant of 24, such as being invertible and, in a geometric sense, scaling volumes by a factor of 24.

Understanding the mathematical underpinnings of determinants enhances our ability to tackle complex problems in various fields. Whether it's solving abstract algebraic equations or analyzing real-world phenomena through mathematical models, the determinant remains an indispensable tool.

Conclusion: The Power of a Single Number

In conclusion, the determinant of a matrix, while seemingly a simple numerical value, holds a wealth of information about the matrix and the linear transformation it represents. We've seen how to calculate it using methods like cofactor expansion and Sarrus' rule, and how to use a given determinant value to solve for unknown variables within the matrix. In our problem, by carefully calculating the determinant of the given 3x3 matrix and setting it equal to 24, we successfully found that the unknown value b must be 8. This exercise not only tests computational skills but also reinforces the understanding that the determinant is a direct consequence of all the elements within the matrix.

The ability to find unknown entries based on determinant properties is a valuable skill in linear algebra, applicable in areas ranging from computer graphics to engineering and economics. It highlights the interconnectedness of matrix elements and their collective impact on the matrix's overall characteristics. So, the next time you encounter a matrix, remember that its determinant is a powerful descriptor, unlocking insights into its invertibility, geometric transformations, and solutions to systems of equations.

For further exploration into the fascinating world of matrices and determinants, you can check out resources like Khan Academy's Linear Algebra section, which offers comprehensive explanations and practice problems. Another excellent resource is Brilliant.org's Linear Algebra course, which provides interactive learning experiences that can deepen your understanding of these fundamental mathematical concepts.