Solve For 'a': Matrix Equation Det(A-B+3C) = -11

Hey everyone, let's dive into a cool matrix problem today, guys! We've got three matrices, A, B, and C, and we need to find a specific value, 'a', that makes a certain equation true. The equation involves the determinant of a combination of these matrices. Sounds complicated? Don't worry, we'll break it down step-by-step. So, grab your calculators, maybe a snack, and let's get this done. Remember, practice makes perfect when it comes to matrices, and understanding determinants is a super key skill in linear algebra. We're going to be looking at:

• Matrix A: $\begin{bmatrix} 4 & 1 \ 3 & 6 \end{bmatrix}$
• Matrix B: $\begin{bmatrix} -1 & 1 \ 4 & -1 \end{bmatrix}$
• Matrix C: $\begin{bmatrix} 3 & -1 \ 1 & -17 \end{bmatrix}$

And the equation we need to satisfy is: det(A - B + 3C) = -11. Our mission, should we choose to accept it, is to find the value of 'a' that makes this equation hold true. Let's get started!

Step 1: Understanding the Matrices and the Goal

Alright team, before we jump into calculations, let's make sure we're all on the same page about what we're dealing with. We've got our matrices A, B, and C. Notice that matrix C has an 'a' in it. Specifically, C = $\begin{bmatrix} 3 & -1 \ 1 & -17a \end{bmatrix}$. This 'a' is our mystery variable, the one we need to solve for. The equation we're given is det(A - B + 3C) = -11. The 'det' stands for determinant, which is a scalar value that can be computed from the elements of a square matrix. It tells us a lot about the matrix, like whether it's invertible. The expression A - B + 3C means we'll be performing matrix subtraction and scalar multiplication. First, we multiply matrix C by 3, then we subtract matrix B from matrix A, and finally, we add the result of (3 * C) to that. Once we have this new resulting matrix, we'll calculate its determinant and set it equal to -11. This will give us a linear equation in terms of 'a', which we can then solve. Pretty straightforward when you break it down, right? It’s all about following the order of operations and knowing how to work with matrices. So, the main goal is to find the value of 'a' that satisfies the given determinant equation. This problem tests our understanding of basic matrix operations and the concept of determinants. Let's keep our focus and work through this methodically. We've got this!

Step 2: Calculating 3C

Okay guys, the first operation we need to perform in the expression A - B + 3C is scalar multiplication. That means we need to multiply each element of matrix C by the scalar value 3. Let's do that:

Matrix C: $\begin{bmatrix} 3 & -1 \ 1 & -17a \end{bmatrix}$

**3C = 3 *** $\begin{bmatrix} 3 & -1 \ 1 & -17a \end{bmatrix}$

To find 3C, we multiply each entry in C by 3:

3 * 3 = 9

3 * (-1) = -3

3 * 1 = 3

3 * (-17a) = -51a

So, our new matrix 3C is:

3C = $\begin{bmatrix} 9 & -3 \ 3 & -51a \end{bmatrix}$

See? Not too bad! We've successfully scaled matrix C. This is a fundamental step in matrix algebra. Remembering to multiply every element by the scalar is crucial. Sometimes, people forget to multiply the variable term, but we've got that covered here with -51a. This scaled matrix will be used in the next step when we combine it with matrices A and B. Keep that result handy, and let's move on to combining the matrices.

Step 3: Calculating A - B + 3C

Now that we have our scaled matrix 3C, it's time to combine A, B, and 3C according to the expression A - B + 3C. Remember, matrix addition and subtraction are done element-wise. This means we add or subtract corresponding elements from each matrix. Let's lay out our matrices:

Matrix A: $\begin{bmatrix} 4 & 1 \ 3 & 6 \end{bmatrix}$

Matrix B: $\begin{bmatrix} -1 & 1 \ 4 & -1 \end{bmatrix}$

Matrix 3C: $\begin{bmatrix} 9 & -3 \ 3 & -51a \end{bmatrix}$

We need to calculate A - B + 3C. Let's do it step by step, or we can combine them all at once. Combining them at once might be faster if we're careful:

New Matrix = $\begin{bmatrix} (4 - (-1) + 9) & (1 - 1 + (-3)) \ (3 - 4 + 3) & (6 - (-1) + (-51a)) \end{bmatrix}$

Let's calculate each element:

• Top-left element: $4 - (-1) + 9 = 4 + 1 + 9 = 14$
• Top-right element: $1 - 1 + (-3) = 0 - 3 = -3$
• Bottom-left element: $3 - 4 + 3 = -1 + 3 = 2$
• Bottom-right element: $6 - (-1) + (-51a) = 6 + 1 - 51a = 7 - 51a$

So, the resulting matrix from A - B + 3C is:

Resulting Matrix = $\begin{bmatrix} 14 & -3 \ 2 & 7 - 51a \end{bmatrix}$

Awesome! We've successfully combined the matrices. This step is all about careful arithmetic. Double-checking each subtraction and addition is super important to avoid errors that could cascade. For example, subtracting a negative number becomes addition, like in $4 - (-1)$, which is $4 + 1$. Also, remember that when combining multiple operations like A - B + 3C, we treat it as (A - B) + 3C, performing the operations sequentially, or simply combine corresponding elements from all matrices at once, ensuring the signs are correct. We're now one step closer to finding 'a' because we have the matrix whose determinant we need to calculate.

Step 4: Calculating the Determinant

We are now at the fun part – calculating the determinant of our resulting matrix! Our matrix is:

Resulting Matrix = $\begin{bmatrix} 14 & -3 \ 2 & 7 - 51a \end{bmatrix}$

For a 2x2 matrix of the form $\begin{bmatrix} p & q \ r & s \end{bmatrix}$, the determinant is calculated as ps - qr. Let's apply this formula to our matrix:

Here, $p = 14$, $q = -3$, $r = 2$, and $s = (7 - 51a)$.

Determinant = $(14) * (7 - 51a) - (-3) * (2)$

Let's expand this:

Determinant = $(14 * 7) - (14 * 51a) - (-6)$

Determinant = $98 - 714a + 6$

Now, let's combine the constant terms:

Determinant = $104 - 714a$

Fantastic! We've calculated the determinant of the combined matrix. This is where the 'a' variable really comes into play. The calculation of the determinant is a critical step and involves applying the correct formula for a 2x2 matrix. The formula ps - qr is straightforward but requires careful multiplication and subtraction. Pay close attention to the signs, especially when you have negative numbers involved, like $-3$ and $-51a$. Expanding the terms correctly, like $14 * (7 - 51a)$, into $98 - 714a$, is also key. Finally, combining the numerical constants $98 + 6$ to get $104$ gives us our final expression for the determinant in terms of 'a'. This expression, 104 - 714a, is what we will set equal to -11 in the next step.

Step 5: Solving for 'a'

We've done all the heavy lifting with the matrices and determinants! Now, we just need to solve for 'a' using the equation given in the problem. We found that the determinant of A - B + 3C is 104 - 714a. The problem states that this determinant must be equal to -11.

So, we set up our equation:

104 - 714a = -11

Our goal now is to isolate 'a'. First, let's subtract 104 from both sides of the equation:

-714a = -11 - 104

-714a = -115

Now, to find 'a', we need to divide both sides by -714:

a = $\frac{-115}{-714}$

Since a negative number divided by a negative number is positive, we get:

a = $\frac{115}{714}$

And there you have it, guys! We've found the value of 'a'. This step is all about basic algebra. Once you have the determinant expressed in terms of 'a', solving for 'a' is just a matter of rearranging the equation. Remember to handle the signs correctly throughout the process. Subtracting 104 from both sides correctly gives -115 on the right. Then, dividing by -714 gives us our final answer. It's always a good idea to check if the fraction can be simplified. In this case, 115 and 714 don't share any common factors other than 1, so the fraction is already in its simplest form. You could also plug this value of 'a' back into the original determinant expression to verify that it indeed equals -11. This problem really showcases how matrix operations and algebraic solving go hand-in-hand to solve complex-looking problems. Great job everyone!

Conclusion

So there you have it! By carefully following the steps of matrix operations – scalar multiplication, matrix addition/subtraction, and determinant calculation – we were able to find the value of 'a' that satisfies the equation det(A - B + 3C) = -11. The value we found is a = 115/714. This problem is a fantastic example of how fundamental concepts in linear algebra, like determinants and matrix arithmetic, can be combined with basic algebraic techniques to solve for unknowns. Keep practicing these types of problems, guys, because the more you work with matrices, the more comfortable and confident you'll become. Remember to always double-check your calculations, especially with signs and multiplication, as one small error can lead to a completely different answer. Whether you're studying for an exam or just expanding your mathematical toolkit, understanding these operations is super valuable. Keep up the great work, and don't hesitate to tackle more challenging matrix problems!