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:

[112713]G=[541497491],\begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix} \cdot G = \begin{bmatrix} \dfrac{-5}{4} & 14 \\ \dfrac{-97}{4} & 91 \end{bmatrix},

where G=[g1g2g3g4]G = \begin{bmatrix} g_1 & g_2 \\ g_3 & g_4 \end{bmatrix}.

To solve for GG, we need to isolate it by multiplying both sides of the equation by the inverse of the matrix A=[112713]A = \begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix}.

Step 1: Find the inverse of AA

The formula for the inverse of a 2x2 matrix A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} is:

A1=1adbc[dbca].A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.

For A=[112713]A = \begin{bmatrix} 11 & 2 \\ 7 & 13 \end{bmatrix}:

  • a=11a = 11, b=2b = 2, c=7c = 7, d=13d = 13,
  • Determinant det(A)=adbc=11×137×2=14314=129\det(A) = ad - bc = 11 \times 13 - 7 \times 2 = 143 - 14 = 129.

Thus, the inverse is:

A1=1129[132711]=[131292129712911129].A^{-1} = \frac{1}{129} \begin{bmatrix} 13 & -2 \\ -7 & 11 \end{bmatrix} = \begin{bmatrix} \frac{13}{129} & \frac{-2}{129} \\ \frac{-7}{129} & \frac{11}{129} \end{bmatrix}.

Step 2: Multiply A1A^{-1} by the matrix on the right-hand side

Now, to find GG, we compute:

G=A1[541497491].G = A^{-1} \cdot \begin{bmatrix} \dfrac{-5}{4} & 14 \\ \dfrac{-97}{4} & 91 \end{bmatrix}.

We will focus on finding g2g_2, which is the element in the first row, second column of the resulting matrix.

g2=(13129)×14+(2129)×91.g_2 = \left( \frac{13}{129} \right) \times 14 + \left( \frac{-2}{129} \right) \times 91.

Step 3: Calculate g2g_2

First, calculate each term:

  • 13129×14=13×14129=182129\frac{13}{129} \times 14 = \frac{13 \times 14}{129} = \frac{182}{129},
  • 2129×91=2×91129=182129\frac{-2}{129} \times 91 = \frac{-2 \times 91}{129} = \frac{-182}{129}.

Now, sum the two terms:

g2=182129+182129=0.g_2 = \frac{182}{129} + \frac{-182}{129} = 0.

Thus, the value of g2g_2 is 0\boxed{0}.


Do you want further details or have any questions about this?

Here are five related questions:

  1. How is the determinant of a 2x2 matrix calculated?
  2. Can all 2x2 matrices be inverted?
  3. What happens if the determinant of a matrix is zero?
  4. How do you multiply two matrices together?
  5. 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