Math Problem Statement
Diberi bahawa matriks D = [[5, 3], [6, 4]] dan E = [[2, -3], [m, n]] serta DE = [[1, 0], [0, 1]]. Cari nilai m dan n.
Solution
Let's solve this problem step by step. The matrices given are:
and the product is given as:
This product represents the identity matrix , so must be the inverse of . Let's calculate and step by step.
Step 1: Compute the determinant of
The determinant of is:
Step 2: Find the inverse of
The inverse of a matrix is: Substitute the values of and :
= \begin{pmatrix} 2 & -\frac{3}{2} \\ -3 & \frac{5}{2} \end{pmatrix}.$$ So, $$E = D^{-1}$$, and we can compare it directly to $$E = \begin{pmatrix} 2 & -3 \\ m & n \end{pmatrix}$$. --- ### Step 3: Equate elements of $$E$$ From the matrices: $$E = \begin{pmatrix} 2 & -3 \\ m & n \end{pmatrix}, \quad D^{-1} = \begin{pmatrix} 2 & -\frac{3}{2} \\ -3 & \frac{5}{2} \end{pmatrix}.$$ Equating the corresponding elements: - $$m = -3$$, - $$n = \frac{5}{2}$$. Thus: $$m = -3, \quad n = \frac{5}{2}.$$ --- Let me know if you'd like further clarification! ### Related Questions: 1. How is the determinant of a $$2 \times 2$$ matrix calculated? 2. What are the properties of the identity matrix $$I$$? 3. Can every square matrix be inverted? Why or why not? 4. How do we check if two matrices are inverses of each other? 5. How do matrix inverses apply in solving systems of linear equations? ### Tip: To verify your results, always multiply the original matrix by its calculated inverse. If the result is the identity matrix, your inverse is correct!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix multiplication
Matrix inverses
Determinants
Identity matrix
Formulas
Matrix multiplication: (AB)_{ij} = sum_k (A_{ik} * B_{kj})
Inverse of a 2x2 matrix: D^{-1} = (1/det(D)) * [[d, -b], [-c, a]]
Determinant of a 2x2 matrix: det(D) = ad - bc
Theorems
Properties of the identity matrix
Invertible matrix theorem
Suitable Grade Level
Grades 10-12