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## Algebra (all content)

### Course: Algebra (all content)>Unit 20

Lesson 16: Solving equations with inverse matrices

# Representing linear systems with matrix equations

Sal shows how a system of two linear equations can be represented with the equation A*x=b where A is the coefficient matrix, x is the variable vector, and b is the constant vector. Created by Sal Khan.

## Want to join the conversation?

• Can you explain the Gaussian elimination?
• At , why does Salman call X and B vectors?
• This is pretty late, but he's just referring to the column. A column vector's just a fancier way of saying that.
• Can someone please help me understand what does the determinant intuitively means...(for instance, the intuition of the first derivative of a function is that it represents the slope of the graph at a given point)..Sal probably has explained this but i cant seem to find the video
• A 2x2 matrix represents a transformation of the plane that keeps the origin fixed and ensures that if two lines were parallel before the transformation, they are still parallel afterward. So rotations and dilations are examples of such transformations, and these transformations will turn a square into a parallelogram.

The determinant is a measure of how much the transformation compresses/stretches the square. If a square has area 1 before the transformation, its area afterward will be equal to the determinant.

The same holds in 3 dimensions; a 3x3 matrix transforms 3d space, and the sides of a cube will stay parallel. If a cube has volume 1 before the transformation, the resulting shape will have volume equal to the determinant.
• Hey, this is a much better video than what you have over here - https://www.khanacademy.org/math/precalculus/precalc-matrices/inverting_matrices/v/matrices-to-solve-a-system-of-equations- would you mind replacing them? :P
• If:
1. We don't know that the matrix equation represents a system of equations.
2. Hence, We treat the matrix equation as a pure math problem. (ie, we don't consider what it represents in order to avoid restricting the range of its solutions)

Question:
Can b*A^-1 be the other solution?
• No, because AbA⁻¹ doesn't necessarily equal b. Matrix multiplication isn't always commutative.
• why didn't we use the row-echelon form?
• what is a column vector
(1 vote)
• A column vector is just a matrix with only one row. They're just another way of writing vectors.
• At why do we multiply left-hand sides of both equations by an inverse of matrix A, why not right-hand sides?