Math Problem Statement
Show that B is the inverse of A. A =
1 2
3 4 , B =
−2 1
3 2 − 1 2 AB =
= I
BA =
= I
Solution
To show that matrix is the inverse of matrix , we need to demonstrate that both and , where is the identity matrix. Here's how we can verify it:
Matrix and :
1 & 2 \\ 3 & 4 \end{pmatrix} , \quad B = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$$ ### Step 1: Calculate $$AB$$ $$AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix}$$ Now, perform the matrix multiplication: $$AB = \begin{pmatrix} (1 \times -2 + 2 \times 1.5) & (1 \times 1 + 2 \times -0.5) \\ (3 \times -2 + 4 \times 1.5) & (3 \times 1 + 4 \times -0.5) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ This shows that $$AB = I$$. ### Step 2: Calculate $$BA$$ $$BA = \begin{pmatrix} -2 & 1 \\ 1.5 & -0.5 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ Now, perform the matrix multiplication: $$BA = \begin{pmatrix} (-2 \times 1 + 1 \times 3) & (-2 \times 2 + 1 \times 4) \\ (1.5 \times 1 + -0.5 \times 3) & (1.5 \times 2 + -0.5 \times 4) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ This shows that $$BA = I$$. ### Conclusion: Since both $$AB = I$$ and $$BA = I$$, we have demonstrated that $$B$$ is indeed the inverse of $$A$$. Would you like a more detailed breakdown of any steps or explanations? --- Here are 5 related questions to explore further: 1. How do you find the inverse of a 2x2 matrix using a formula? 2. What conditions must a matrix meet to have an inverse? 3. Can every square matrix have an inverse? 4. What is the determinant of matrix $$A$$ and why is it important for inverses? 5. How does matrix inversion work for larger matrices (e.g., 3x3)? **Tip:** For a 2x2 matrix $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, the inverse exists if $$ad - bc \neq 0$$. The inverse formula is $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Matrix Inversion
Linear Algebra
Formulas
Matrix multiplication: (AB)_{ij} = Σ A_{ik} B_{kj}
Inverse of a 2x2 matrix A = [[a, b], [c, d]]: A^{-1} = (1/det(A)) * [[d, -b], [-c, a]]
Determinant of a 2x2 matrix: det(A) = ad - bc
Theorems
Properties of Inverse Matrices: AB = BA = I
Conditions for Matrix Inverses: det(A) ≠ 0
Suitable Grade Level
Grades 10-12