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Step 1:I'll solve this step-by-step, using precise LaTeX formatting:
Step 2:: Understanding the Discriminant
\Delta = ad - bc
The discriminant for a 2×2 matrix is a scalar value that provides important information about the matrix's characteristics.
Step 3:: Breaking Down the Components
- $$d$$ is the element in the second row, second column
- b is the element in the first row, second column
Step 4:: Calculation Method
To find the discriminant:
Step 5:
Multiply the elements on the main diagonal ($$a$$ and $$d$$)
Step 6:
Multiply the elements on the other diagonal ($$b$$ and $$c$$)
Step 7:
Subtract the second product from the first product
Step 8:: Significance of the Discriminant
- If $$\Delta < 0$$, the matrix has complex conjugate eigenvalues
- If \Delta = 0, the matrix has repeated eigenvalues
Final Answer
The discriminant of a 2×2 matrix A = \begin{pmatrix} a & b \ c & d \end{pmatrix} is calculated as \Delta = ad - bc.
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