Determinant Of Matrix C Where AC = B

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Let's dive into a matrix problem where we need to find the determinant of matrix C, given matrices A and B and the relationship AC = B. This is a classic linear algebra problem, and we'll break it down step by step. So, buckle up, guys, it's gonna be a fun ride!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the problem is asking. We have three matrices: A, B, and C. We know A and B, and we know that when we multiply A by C, we get B. Our mission, should we choose to accept it, is to find the determinant of C. Remember, the determinant is a special number that can be calculated from a square matrix, and it tells us a lot about the matrix. It's like the matrix's secret identity! The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. Understanding the concept of determinants is crucial for grasping the invertibility of matrices and solving systems of linear equations. So, let's make sure we're solid on this foundation. We'll need to use properties of determinants and matrix multiplication to solve this problem efficiently. And don't worry, we'll take it slow and explain each step along the way, so you can follow along even if you're not a matrix guru. This problem isn't just about crunching numbers; it's about understanding the relationships between matrices and their determinants. It's about seeing how matrix multiplication affects determinants and how we can use that knowledge to solve for unknowns. So, stay focused, and let's get started!

Key Concepts and Formulas

Before we start crunching numbers, let's review some key concepts and formulas that will help us solve this problem. These are the tools we'll need in our mathematical toolbox. First, we need to remember how to calculate the determinant of a 2x2 matrix. If we have a matrix like this:

(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}

Then the determinant is calculated as:

det = ad - bc

This formula is essential, so make sure you have it memorized. It's the foundation for many matrix calculations. Next, we need to understand how determinants behave with matrix multiplication. This is a crucial property that we'll use to solve the problem. The property states that if we have two matrices, let's say A and C, then the determinant of their product is the product of their determinants:

det(AC) = det(A) * det(C)

This is a powerful tool because it allows us to relate the determinant of the product of two matrices to the individual determinants of the matrices. It's like saying that the "determinant" of doing two things in a row is the same as doing each thing separately and then multiplying the results. Also, remember that for a matrix to have an inverse, its determinant must not be zero. If the determinant is zero, the matrix is singular and does not have an inverse. This is a critical point to keep in mind when working with matrices. Finally, we need to remember the basic rules of matrix multiplication. We won't be explicitly multiplying matrices in this problem, but understanding how matrix multiplication works is essential for understanding the relationship AC = B. Basically, when we multiply two matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix. Mastering these concepts is crucial for tackling matrix problems effectively. With these tools in our toolbox, we're ready to tackle the problem at hand. Let's move on to the solution!

Solution Steps

Okay, let's get down to business and solve this problem step by step. Remember, we want to find the determinant of matrix C, and we know that AC = B. Here's how we can do it: First, let's find the determinant of matrix A. Matrix A is given as:

A=(1125)A = \begin{pmatrix} 1 & 1 \\ 2 & 5 \end{pmatrix}

Using the formula for the determinant of a 2x2 matrix, we have:

det(A) = (1 * 5) - (1 * 2) = 5 - 2 = 3

So, the determinant of A is 3. Next, let's find the determinant of matrix B. Matrix B is given as:

B=(23−13)B = \begin{pmatrix} 2 & 3 \\ -1 & 3 \end{pmatrix}

Using the same formula, we have:

det(B) = (2 * 3) - (3 * -1) = 6 + 3 = 9

So, the determinant of B is 9. Now, remember the property that det(AC) = det(A) * det(C). Since AC = B, we can write:

det(B) = det(A) * det(C)

We know det(A) = 3 and det(B) = 9, so we can substitute these values into the equation:

9 = 3 * det(C)

Now, we can solve for det(C) by dividing both sides of the equation by 3:

det(C) = 9 / 3 = 3

Therefore, the determinant of matrix C is 3. And that's it! We've solved the problem. By using the properties of determinants and matrix multiplication, we were able to find the determinant of C without explicitly finding the matrix C itself. Wasn't that fun? Let's recap the steps we took:

  1. Found the determinant of matrix A.
  2. Found the determinant of matrix B.
  3. Used the property det(AC) = det(A) * det(C) to relate the determinants.
  4. Solved for det(C).

Answer

The determinant of C is 3, so the answer is (b). Great job, everyone! You've successfully navigated through this matrix problem. Remember to practice these concepts to solidify your understanding. And now you can confidently tell your friends that you know how to find the determinant of a matrix when given the product of two matrices. Keep up the great work, and I'll see you in the next math adventure!