Transcribed Image Text:Evaluate the determinant.
\[
\begin{vmatrix}
\cos(\theta) & 1 \\
1 & \cos(\theta)
\end{vmatrix}
\]
In this problem, you need to find the determinant of a 2x2 matrix with elements \(\cos(\theta)\) and 1.
**Explanation of the Matrix:**
- The matrix is a square matrix with two rows and two columns.
- The first row contains the elements \(\cos(\theta)\) and 1.
- The second row contains the elements 1 and \(\cos(\theta)\).
**Calculating the Determinant:**
For a 2x2 matrix of the form:
\[
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\]
The determinant is calculated as:
\[ ad - bc \]
Applying this to our matrix:
\[ (\cos(\theta) \times \cos(\theta)) - (1 \times 1) \]
This simplifies to:
\[ \cos^2(\theta) - 1 \]
The text box and red X symbol indicate an input area for the answer and mark where an incorrect attempt might be displayed, respectively.
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
![Evaluate the determinant.
\[
\begin{vmatrix}
\cos(\theta) & 1 \\
1 & \cos(\theta)
\end{vmatrix}
\]
In this problem, you need to find the determinant of a 2x2 matrix with elements \(\cos(\theta)\) and 1.
**Explanation of the Matrix:**
- The matrix is a square matrix with two rows and two columns.
- The first row contains the elements \(\cos(\theta)\) and 1.
- The second row contains the elements 1 and \(\cos(\theta)\).
**Calculating the Determinant:**
For a 2x2 matrix of the form:
\[
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\]
The determinant is calculated as:
\[ ad - bc \]
Applying this to our matrix:
\[ (\cos(\theta) \times \cos(\theta)) - (1 \times 1) \]
This simplifies to:
\[ \cos^2(\theta) - 1 \]
The text box and red X symbol indicate an input area for the answer and mark where an incorrect attempt might be displayed, respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc42b80bf-a5d4-414b-bce1-0fe52a04dbbd%2F2276ee43-14ee-4b43-96fe-8d35db21fd86%2F5pou7js_processed.png&w=3840&q=75)
