In Exercises 17-26,let A=[ -3 -7 2 -9 5 0 ] and B=[ -5 -1 0

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In Exercises 17-26,let A=[ -3 -7 2 -9 5 ...

In Exercises $17-26,$let
$$A=\left[\begin{array}{rr}{-3} & {-7} \\ {2} & {-9} \\ {5} & {0}\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{rr}{-5} & {-1} \\ {0} & {0} \\ {3} & {-4}\end{array}\right]$$
Solve each matrix equation for $X .$
$$
2 X+5 A=B
$$

Key Concepts

Matrix Equation

A matrix equation is an equality involving matrices, where the operations of addition, subtraction, and scalar multiplication are performed element-wise. Solving a matrix equation involves performing equivalent operations on both sides of the equation while respecting the properties of matrices, such as ensuring that matrices being added or subtracted are of the same dimensions.

Scalar Multiplication

Scalar multiplication in the context of matrices involves multiplying every entry in the matrix by a scalar (a real or complex number). This operation is essential in manipulating terms in a matrix equation, such as isolating the unknown matrix through division by a scalar, assuming the scalar is nonzero.

Matrix Addition and Subtraction

Matrix addition and subtraction are fundamental operations where the corresponding entries of matrices of the same dimension are added or subtracted. These operations obey the associative and commutative properties, allowing for systematic isolation of the unknown matrix in an equation by adding or subtracting known matrices.

Isolating the Unknown Matrix

In solving matrix equations, the goal is to isolate the unknown matrix on one side of the equation. This process involves using inverse operations, such as subtracting a known matrix from both sides or performing scalar division (multiplying by the reciprocal of a nonzero scalar) to remove coefficients attached to the unknown matrix. This is analogous to solving linear equations with real numbers.

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