Let A=[−3−72−950]A = \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix} and B=[−5−1003−4]B = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix}. Solve each matrix equation for X. 2X + A = B

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 19

7장, 문제 19

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. 2X + A = B

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Identify the given matrices and the equation to solve: $2X + A = B$, where $A = \begin{bmatrix} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{bmatrix}$.

Isolate the term with $X$ by subtracting matrix $A$ from both sides of the equation: $2X = B - A$.

Perform the matrix subtraction $B - A$ by subtracting corresponding elements of $A$ from $B$: $\left(b_{ij} - a_{ij}\right)$ for each element.

Divide each element of the resulting matrix \$2X$ by 2 to solve for $X$: $X = \frac{1}{2}(B - A)$.

Write the final expression for $X$ as $X = \frac{1}{2} \left( B - A \right)$, which represents the solution matrix.

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주요 개념

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Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same dimensions. Each element in the resulting matrix is the sum or difference of the elements in the same position from the original matrices. This operation is fundamental for manipulating matrix equations like 2X + A = B.

Scalar Multiplication of Matrices

Scalar multiplication involves multiplying every element of a matrix by a constant (scalar). For example, multiplying matrix X by 2 means doubling each element of X. This operation is essential for isolating the matrix variable in equations such as 2X + A = B.

Solving Matrix Equations

To solve matrix equations like 2X + A = B, you isolate the matrix variable by performing inverse operations, such as subtracting A from both sides and then dividing by the scalar. Understanding how to manipulate matrices algebraically is key to finding the unknown matrix X.

Let A=[−3−72−950]A = \(\begin{bmatrix}\) -3 & -7 \\ 2 & -9 \\ 5 & 0 \(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\) -5 & -1 \\ 0 & 0 \\ 3 & -4 \(\end{bmatrix}\). Solve each matrix equation for X. 2X + A = B