Math Problem Statement
If A=[8511−4]A=[8115−4] and B=[−4562]B=[−4652] then |4A−3B−5I|=?
Solution
We are tasked to compute , where , , and are given matrices. Here's the step-by-step solution:
Step 1: Write Down the Matrices
Matrix is:
\begin{bmatrix} 8 & 5 \\ 1 & -4 \end{bmatrix}$$ Matrix $$B$$ is: $$B = \begin{bmatrix} -4 & 5 \\ 6 & 2 \end{bmatrix}$$ Matrix $$I$$ is the $$2 \times 2$$ identity matrix: $$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ ### Step 2: Compute $$4A$$ Multiply each entry of $$A$$ by 4: $$4A = 4 \cdot \begin{bmatrix} 8 & 5 \\ 1 & -4 \end{bmatrix} = \begin{bmatrix} 32 & 20 \\ 4 & -16 \end{bmatrix}$$ ### Step 3: Compute $$3B$$ Multiply each entry of $$B$$ by 3: $$3B = 3 \cdot \begin{bmatrix} -4 & 5 \\ 6 & 2 \end{bmatrix} = \begin{bmatrix} -12 & 15 \\ 18 & 6 \end{bmatrix}$$ ### Step 4: Compute $$5I$$ Multiply each entry of $$I$$ by 5: $$5I = 5 \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$$ ### Step 5: Compute $$4A - 3B - 5I$$ Perform matrix subtraction: $$4A - 3B - 5I = \begin{bmatrix} 32 & 20 \\ 4 & -16 \end{bmatrix} - \begin{bmatrix} -12 & 15 \\ 18 & 6 \end{bmatrix} - \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$$ Step-by-step: 1. First compute $$4A - 3B$$: $$4A - 3B = \begin{bmatrix} 32 - (-12) & 20 - 15 \\ 4 - 18 & -16 - 6 \end{bmatrix} = \begin{bmatrix} 44 & 5 \\ -14 & -22 \end{bmatrix}$$ 2. Now subtract $$5I$$: $$4A - 3B - 5I = \begin{bmatrix} 44 - 5 & 5 - 0 \\ -14 - 0 & -22 - 5 \end{bmatrix} = \begin{bmatrix} 39 & 5 \\ -14 & -27 \end{bmatrix}$$ ### Step 6: Compute the Determinant of $$4A - 3B - 5I$$ The determinant of a $$2 \times 2$$ matrix $$M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ is: $$|M| = ad - bc$$ For the matrix: $$\begin{bmatrix} 39 & 5 \\ -14 & -27 \end{bmatrix}$$ The determinant is: $$|4A - 3B - 5I| = (39)(-27) - (5)(-14) = -1053 + 70 = -983$$ ### Final Answer: $$|4A - 3B - 5I| = -983$$ --- Do you have any questions or need more explanation? ### Related Questions: 1. How do you compute the determinant of larger matrices (e.g., $$3 \times 3$$)? 2. What are the properties of the identity matrix and their significance in matrix operations? 3. Can determinants be negative, and what does it mean if they are? 4. How does matrix scalar multiplication affect its determinant? 5. How can the determinant help in solving systems of linear equations? ### Tip: Always simplify matrix operations step-by-step to avoid errors, especially in subtraction and determinant calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Determinants
Matrix Multiplication
Matrix Subtraction
Identity Matrix
Formulas
Determinant of a 2x2 matrix: |M| = ad - bc for M = [[a, b], [c, d]]
Matrix scalar multiplication: k * A = [[k*a11, k*a12], [k*a21, k*a22]]
Theorems
Properties of Determinants
Properties of the Identity Matrix in Matrix Operations
Suitable Grade Level
Grades 10-12