Math Problem Statement
If \begin{bmatrix} 11&2\ 7&13 \end{bmatrix}\cdot G= \begin{bmatrix} \dfrac{-5}{4}&14\ \dfrac{-97}{4}&91 \end{bmatrix}, then G=\begin{bmatrix} g_{1}&g_{2}\ g_{3}&g_{4} \end{bmatrix}.
What is the value of g_{2}?
(It's okay to use a calculator to find the necessary inverse and multiply.)
Solution
We are given the matrix equation:
where .
To solve for , we need to isolate it by multiplying both sides of the equation by the inverse of the matrix .
Step 1: Find the inverse of
The formula for the inverse of a 2x2 matrix is:
For :
- , , , ,
- Determinant .
Thus, the inverse is:
Step 2: Multiply by the matrix on the right-hand side
Now, to find , we compute:
We will focus on finding , which is the element in the first row, second column of the resulting matrix.
Step 3: Calculate
First, calculate each term:
- ,
- .
Now, sum the two terms:
Thus, the value of is .
Do you want further details or have any questions about this?
Here are five related questions:
- How is the determinant of a 2x2 matrix calculated?
- Can all 2x2 matrices be inverted?
- What happens if the determinant of a matrix is zero?
- How do you multiply two matrices together?
- What are some properties of matrix inverses?
Tip: Always check that the determinant is non-zero before attempting to find the inverse of a matrix!
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Multiplication
Matrix Inverse
Formulas
Inverse of a 2x2 matrix: \( A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)
Matrix multiplication: \( C = A \cdot B \)
Theorems
Inverse Matrix Theorem
Suitable Grade Level
College-level Linear Algebra