Determinant Of Symmetric Matrix A: Step-by-Step Solution

Oct 17, 2025 by ADMIN 57 views

Hey guys! Today, we're diving into a cool problem in linear algebra: finding the determinant of a symmetric matrix. Specifically, we're given a matrix A and told it's symmetric. Our mission, should we choose to accept it, is to find its determinant. Let's break it down and make sure we understand every step. So, if you're ready to unravel this matrix mystery, stick around, and let’s get started!

Understanding Symmetric Matrices

Before we jump into the calculation, let's quickly recap what a symmetric matrix actually is. A matrix A is symmetric if it's equal to its transpose. In simpler terms, if you flip the matrix over its main diagonal (the diagonal from the top-left to the bottom-right), you get the same matrix back. Mathematically, this means A = A^T. This property is super important because it gives us some crucial clues about the elements of our matrix.

For our given matrix:

A =  egin{pmatrix}
    1 & a-5 & -2 \ 
    2 & 3 & b \ 
    b-a & a-x & 4
    
\end{pmatrix}

If A is symmetric, then:

The element in the first row, second column (a-5) must be equal to the element in the second row, first column (2).
The element in the first row, third column (-2) must be equal to the element in the third row, first column (b-a).
The element in the second row, third column (b) must be equal to the element in the third row, second column (a-x).

These equalities will be key to figuring out the values of a, b, and x. Remember, understanding the properties of symmetric matrices is fundamental to solving this type of problem. It’s like having a secret decoder ring – it unlocks the relationships between the matrix elements!

Solving for the Unknown Variables

Okay, now that we know what makes a matrix symmetric, let's use that knowledge to find the values of a, b, and x in our matrix. Remember those equalities we talked about? They're going to be our best friends here. Let’s go through them one by one.

First up, we have a - 5 = 2. This one's pretty straightforward. To solve for a, we just add 5 to both sides of the equation:

a = 2 + 5 = 7

So, we've found that a equals 7. Awesome! Now, let's move on to the next equality:

-2 = b - a

We already know that a = 7, so we can substitute that into the equation:

-2 = b - 7

To solve for b, we add 7 to both sides:

b = -2 + 7 = 5

Fantastic, we've got b = 5! Only one more variable to go. Let's tackle the last equality:

b = a - x

We know a = 7 and b = 5, so we can substitute those in:

5 = 7 - x

To solve for x, we can rearrange the equation. Subtract 7 from both sides:

5 - 7 = -x

-2 = -x

Multiply both sides by -1:

x = 2

Alright! We've successfully found the values of all three variables: a = 7, b = 5, and x = 2. This is a huge step because now we can plug these values back into our original matrix and see what it actually looks like. It’s like assembling the pieces of a puzzle – we're getting closer to the final picture!

Constructing the Symmetric Matrix

Now that we've cracked the code and found the values of a, b, and x, it's time to put them back into the matrix A. This will give us the concrete symmetric matrix we need to work with. Let's take it step by step. Our original matrix was:

A =  egin{pmatrix}
    1 & a-5 & -2 \ 
    2 & 3 & b \ 
    b-a & a-x & 4
    
\end{pmatrix}

We found that a = 7, b = 5, and x = 2. Let's substitute these values into the matrix:

a - 5 = 7 - 5 = 2
b = 5
b - a = 5 - 7 = -2
a - x = 7 - 2 = 5

Plugging these in, we get:

A =  egin{pmatrix}
    1 & 2 & -2 \ 
    2 & 3 & 5 \ 
    -2 & 5 & 4
    
\end{pmatrix}

Take a moment to double-check that this matrix is indeed symmetric. Look at the elements across the main diagonal. Is the element in the ith row and jth column the same as the element in the jth row and ith column? Yep, it checks out! We've successfully constructed our symmetric matrix. This is a crucial step because now we have the actual numbers we need to calculate the determinant. It's like having the ingredients ready – now we can finally bake the cake (or, in this case, calculate the determinant!).

Calculating the Determinant

Alright, guys, the moment we've been waiting for! Now that we have our symmetric matrix, we can finally calculate its determinant. There are several ways to do this, but for a 3x3 matrix, a common method is using the rule of Sarrus or cofactor expansion. Let’s use cofactor expansion along the first row. Our matrix A is:

A =  egin{pmatrix}
    1 & 2 & -2 \ 
    2 & 3 & 5 \ 
    -2 & 5 & 4
    
\end{pmatrix}

The determinant of A, denoted as |A|, can be calculated as follows:

|A| = 1 * C₁₁ + 2 * C₁₂ + (-2) * C₁₃

Where C_ij represents the cofactor of the element in the ith row and jth column. Remember, the cofactor is calculated as:

C_ij = (-1)^i+j * M_ij

Where M_ij is the minor, which is the determinant of the submatrix formed by deleting the ith row and jth column.

Let's calculate the minors and cofactors:

M₁₁: Delete the first row and first column:
```
egin{pmatrix}
    3 & 5 \ 
    5 & 4
```

\end{pmatrix} ```

M<sub>11</sub> = (3 * 4) - (5 * 5) = 12 - 25 = -13
C<sub>11</sub> = (-1)<sup>1+1</sup> * (-13) = 1 * -13 = -13

M₁₂: Delete the first row and second column:
```
egin{pmatrix}
    2 & 5 \ 
    -2 & 4
```

\end{pmatrix} ```

M<sub>12</sub> = (2 * 4) - (5 * -2) = 8 + 10 = 18
C<sub>12</sub> = (-1)<sup>1+2</sup> * 18 = -1 * 18 = -18

M₁₃: Delete the first row and third column:
```
egin{pmatrix}
    2 & 3 \ 
    -2 & 5
```

\end{pmatrix} ```

M<sub>13</sub> = (2 * 5) - (3 * -2) = 10 + 6 = 16
C<sub>13</sub> = (-1)<sup>1+3</sup> * 16 = 1 * 16 = 16

Now, plug these cofactors back into the determinant formula:

|A| = 1 * (-13) + 2 * (-18) + (-2) * 16 |A| = -13 - 36 - 32 |A| = -81

So, the determinant of matrix A is -81! We did it! This was a bit of a journey, but we made it through the cofactor expansion and found our answer. Remember, practice makes perfect, so the more determinants you calculate, the easier it will become.

Conclusion

And there you have it! We've successfully navigated through the process of finding the determinant of a symmetric matrix. We started by understanding the properties of symmetric matrices, then solved for the unknown variables, constructed the actual matrix, and finally, calculated the determinant using cofactor expansion. The answer, as we found, is -81. This problem highlights the importance of understanding the underlying concepts and properties in linear algebra. Knowing that a matrix is symmetric gives us valuable information that simplifies the problem. So, keep practicing, keep exploring, and you'll become a matrix-solving pro in no time! If you guys have any questions or want to dive deeper into linear algebra, drop a comment below. Let's keep the learning going!