Solve AX=B With Determinants: A Step-by-Step Guide

Hey guys! Today, we're diving into how to solve a linear system of equations using determinants, specifically focusing on the system AX = B. We'll break down each step to make sure you get a solid understanding of the process. Let's get started!

Understanding the Linear System AX = B

First, let's clarify what we mean by the linear system AX = B. In this context:

• A is a matrix of coefficients.
• X is a column matrix of variables (unknowns).
• B is a column matrix of constants.

For our specific problem, we have:

• A = $\begin{pmatrix} 1 & 0 & -3 \\ 2 & 3 & -1 \\ 4 & 5 & -2 \end{pmatrix}$
• X = $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$
• B = $\begin{pmatrix} -5 \\ 7 \\ 10 \end{pmatrix}$

So, our system of equations is:

• 1x + 0y - 3z = -5
• 2x + 3y - 1z = 7
• 4x + 5y - 2z = 10

Now that we understand the system, let's move on to solving it using determinants.

Cramer's Rule: The Key to Solving with Determinants

To solve this system using determinants, we'll employ Cramer's Rule. Cramer's Rule provides a method to find the values of x, y, and z by calculating determinants of matrices derived from A and B. The formulas are as follows:

• x = det(A_x) / det(A)
• y = det(A_y) / det(A)
• z = det(A_z) / det(A)

Where:

• det(A) is the determinant of the coefficient matrix A.
• A_x is the matrix formed by replacing the first column of A with the column matrix B.
• A_y is the matrix formed by replacing the second column of A with the column matrix B.
• A_z is the matrix formed by replacing the third column of A with the column matrix B.

Step 1: Calculate det(A)

First, we need to find the determinant of matrix A:

A = $\begin{pmatrix} 1 & 0 & -3 \\ 2 & 3 & -1 \\ 4 & 5 & -2 \end{pmatrix}$

We calculate the determinant as follows:

det(A) = 1 * (3*(-2) - (-1)5) - 0 * (2(-2) - (-1)4) + (-3) * (25 - 3*4)

det(A) = 1 * (-6 + 5) - 0 + (-3) * (10 - 12)

det(A) = 1 * (-1) - 3 * (-2)

det(A) = -1 + 6

det(A) = 5

So, the determinant of A is 5. It's super important to get this right, or everything else will be off! Take your time and double-check those calculations, folks. Trust me, a little extra attention here saves a lot of headaches later.

Step 2: Calculate det(A_x)

Next, we replace the first column of A with B to get A_x:

A_x = $\begin{pmatrix} -5 & 0 & -3 \\ 7 & 3 & -1 \\ 10 & 5 & -2 \end{pmatrix}$

Now, we find the determinant of A_x:

det(A_x) = -5 * (3*(-2) - (-1)5) - 0 * (7(-2) - (-1)10) + (-3) * (75 - 3*10)

det(A_x) = -5 * (-6 + 5) - 0 + (-3) * (35 - 30)

det(A_x) = -5 * (-1) - 3 * (5)

det(A_x) = 5 - 15

det(A_x) = -10

Thus, the determinant of A_x is -10.

Step 3: Calculate det(A_y)

Now, we replace the second column of A with B to get A_y:

A_y = $\begin{pmatrix} 1 & -5 & -3 \\ 2 & 7 & -1 \\ 4 & 10 & -2 \end{pmatrix}$

We calculate the determinant of A_y:

det(A_y) = 1 * (7*(-2) - (-1)10) - (-5) * (2(-2) - (-1)4) + (-3) * (210 - 7*4)

det(A_y) = 1 * (-14 + 10) + 5 * (-4 + 4) + (-3) * (20 - 28)

det(A_y) = 1 * (-4) + 5 * (0) - 3 * (-8)

det(A_y) = -4 + 0 + 24

det(A_y) = 20

So, the determinant of A_y is 20.

Step 4: Calculate det(A_z)

Replace the third column of A with B to get A_z:

A_z = $\begin{pmatrix} 1 & 0 & -5 \\ 2 & 3 & 7 \\ 4 & 5 & 10 \end{pmatrix}$

Now, calculate the determinant of A_z:

det(A_z) = 1 * (310 - 75) - 0 * (210 - 74) + (-5) * (25 - 34)

det(A_z) = 1 * (30 - 35) - 0 + (-5) * (10 - 12)

det(A_z) = 1 * (-5) - 5 * (-2)

det(A_z) = -5 + 10

det(A_z) = 5

Thus, the determinant of A_z is 5. Remember, stay organized when calculating these determinants. It's easy to make a small mistake that throws everything off. Keep track of your signs and terms!

Step 5: Apply Cramer's Rule to Find x, y, and z

Now that we have all the determinants, we can find the values of x, y, and z using Cramer's Rule:

• x = det(A_x) / det(A) = -10 / 5 = -2
• y = det(A_y) / det(A) = 20 / 5 = 4
• z = det(A_z) / det(A) = 5 / 5 = 1

Therefore, the solution to the system is x = -2, y = 4, and z = 1. Boom! We solved it! It's always a good idea to plug these values back into your original equations to make sure they work. Let's do that now.

Verification

Let's plug these values back into the original equations to verify our solution:

1. 1x + 0y - 3z = -5 1*(-2) + 0*(4) - 3*(1) = -2 + 0 - 3 = -5 (Correct!)
2. 2x + 3y - 1z = 7 2*(-2) + 3*(4) - 1*(1) = -4 + 12 - 1 = 7 (Correct!)
3. 4x + 5y - 2z = 10 4*(-2) + 5*(4) - 2*(1) = -8 + 20 - 2 = 10 (Correct!)

Since all three equations hold true, our solution x = -2, y = 4, and z = 1 is correct.

Conclusion

Alright, there you have it! We successfully solved the linear system AX = B using determinants and Cramer's Rule. Remember to take your time, double-check your calculations, and stay organized. This method is powerful, but it relies on accurate determinant calculations. So go ahead, practice this method with different systems, and you'll become a pro in no time! Solving linear systems might seem daunting at first, but with practice and a clear understanding of the steps, you'll be able to tackle any problem that comes your way. Keep up the great work!