Math Problem Statement

Find the inverse of the matrix A = [[2, 3], [3, 5]].

Solution

You are asked to find the inverse of the matrix AA:

A=[2335]A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}

To find the inverse of a 2x2 matrix, we use the formula:

A1=1det(A)[dbca]A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Where the matrix AA is:

A=[abcd]A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

For the given matrix:

A=[2335],a=2,b=3,c=3,d=5A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \quad a = 2, b = 3, c = 3, d = 5

Step 1: Compute the determinant of AA:

det(A)=adbc=(2)(5)(3)(3)=109=1\text{det}(A) = ad - bc = (2)(5) - (3)(3) = 10 - 9 = 1

Step 2: Find the inverse matrix:

Using the formula, the inverse is:

A1=11[5332]A^{-1} = \frac{1}{1} \begin{bmatrix} 5 & -3 \\ -3 & 2 \end{bmatrix}

Thus, the inverse of matrix AA is:

A1=[5332]A^{-1} = \begin{bmatrix} 5 & -3 \\ -3 & 2 \end{bmatrix}

Let me know if you'd like further details, or have any questions. Here are some related questions you might find helpful:

  1. What are the properties of the inverse of a matrix?
  2. Can a matrix always have an inverse?
  3. How do you find the inverse of larger matrices (e.g., 3x3 matrices)?
  4. What is the determinant of a matrix and its significance?
  5. How is the inverse of a matrix used in solving systems of linear equations?

Tip: The inverse of a matrix exists only if its determinant is non-zero. If the determinant is zero, the matrix is called singular, and no inverse exists.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Determinants

Formulas

Inverse of 2x2 matrix formula: A^(-1) = 1/det(A) * [[d, -b], [-c, a]]
Determinant formula: det(A) = ad - bc

Theorems

The Inverse Matrix Theorem
Determinant and Invertibility

Suitable Grade Level

Grades 10-12