Find Matrix A in a System of Equations Using Matrix Multiplication

Math Problem Statement

Find the matrix A such that A * [[1, 0], [-1, 3]] = [[-4, -3], [-1, 15]].

The problem asks us to find the matrix $A$ , such that:

$A \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} -4 & -3 \\ -1 & 15 \end{bmatrix}$

We are given the hint that matrix $A$ has the form:

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

We need to determine the values of $a$ , $b$ , $c$ , and $d$ .

We will multiply matrix $A$ by the matrix on the left-hand side:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ -1 & 3 \end{bmatrix}$

The result of this multiplication is:

= \begin{bmatrix} a - b & 3b \\ c - d & 3d \end{bmatrix}$$ ### Step 2: Set Equal to the Given Matrix We now set this result equal to the right-hand side matrix: $$\begin{bmatrix} a - b & 3b \\ c - d & 3d \end{bmatrix} = \begin{bmatrix} -4 & -3 \\ -1 & 15 \end{bmatrix}$$ ### Step 3: Solve the System of Equations This gives us a system of equations by comparing corresponding elements: 1. $$a - b = -4$$ 2. $$3b = -3$$ 3. $$c - d = -1$$ 4. $$3d = 15$$ ### Step 4: Solve for $$b$$ and $$d$$ From equation (2): $$3b = -3 \implies b = -1$$ From equation (4): $$3d = 15 \implies d = 5$$ ### Step 5: Solve for $$a$$ and $$c$$ Substitute $$b = -1$$ into equation (1): $$a - (-1) = -4 \implies a + 1 = -4 \implies a = -5$$ Substitute $$d = 5$$ into equation (3): $$c - 5 = -1 \implies c = 4$$ ### Final Solution Thus, the matrix $$A$$ is: $$A = \begin{bmatrix} -5 & -1 \\ 4 & 5 \end{bmatrix}$$ Would you like further details or have any questions? --- Here are 5 related questions: 1. How would you solve this if matrix $$A$$ was a 3x3 matrix instead? 2. What is the significance of matrix multiplication in this context? 3. Can you find the inverse of the matrix $$A$$ we calculated? 4. How would you solve this system using determinants? 5. How do these types of problems relate to systems of linear equations? **Tip:** When multiplying matrices, always keep the order of multiplication in mind—matrix multiplication is not commutative!

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Linear Algebra
Matrix Multiplication
Systems of Equations

Matrix multiplication formula

Equality of matrices

College-level or Advanced High School (Grades 11-12)

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