Math Problem Statement

El valor del K que hace que el determinante de la matriz sea 5:

Solution

To solve the problem, we need to find the value of kk that makes the determinant of the given 2×22 \times 2 matrix equal to 5. The matrix is:

3 & k \\ 1 & 5 \end{pmatrix}$$ The determinant of a $$2 \times 2$$ matrix is given by the formula: $$\text{determinant} = (a \cdot d) - (b \cdot c)$$ For this matrix: $$\text{determinant} = (3 \cdot 5) - (k \cdot 1)$$ Simplifying the equation: $$\text{determinant} = 15 - k$$ We are told that the determinant is equal to 5, so: $$15 - k = 5$$ Now, solve for $$k$$: $$15 - 5 = k$$ $$k = 10$$ Thus, the value of $$k$$ that makes the determinant equal to 5 is $$k = 10$$. --- Would you like a more detailed explanation of the steps, or have any other questions? Here are 5 related questions to further your understanding: 1. What happens to the determinant if $$k = 0$$ in this matrix? 2. Can you find the inverse of this matrix when $$k = 10$$? 3. How would the result change if the matrix was $$3 \times 3$$? 4. What are the properties of a matrix with a determinant of 0? 5. How do we find the determinant of a matrix larger than $$2 \times 2$$? **Tip**: In a $$2 \times 2$$ matrix, swapping two rows or two columns changes the sign of the determinant.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Determinants

Formulas

Determinant formula for a 2x2 matrix: det(A) = ad - bc

Theorems

Determinant Theorem for 2x2 matrices

Suitable Grade Level

High School (Grades 9-12)