Let A=[ 1 2 -3 5 ], X=[ x1 x2 ], and B=[ -4 12 ] . Show that the equ

step 1 For this problem, we have been given three matrices, A, X, and B. First, let's take a look at th

Let A=[ 1 2 -3 5 ], X=[ x1 x2 ], and...

Let $A=\left[\begin{array}{rl}{1} & {2} \\ {-3} & {5}\end{array}\right], X=\left[\begin{array}{l}{x_{1}} \\ {x_{2}}\end{array}\right],$ and $B=\left[\begin{array}{c}{-4} \\ {12}\end{array}\right] .$ Show that the equation $A X=B$ represents a linear system of two equations in two unknowns. Solve the system and substitute into the matrix equation to check your results.

Key Concepts

Matrix Representation of Linear Systems

This concept involves expressing a system of linear equations in a compact form using matrices. In a matrix equation like AX = B, the matrix A contains the coefficients of the variables, the column vector X represents the unknowns, and the vector B contains the constant terms. This representation is central in linear algebra as it provides a unified way to deal with systems of equations, aids in understanding the structure of solutions, and facilitates the use of matrix operations to solve the system.

Solving Systems of Linear Equations

This key idea covers methods and techniques for finding the set of values that satisfy every equation in a system. Traditional methods include substitution, elimination, and using matrix techniques such as finding an inverse or employing row-reduction to echelon forms. Solving the system involves finding a unique solution, infinitely many solutions, or determining that no solution exists, based on the properties of the coefficient matrix and the consistency of the system.

Verification of Solutions

After obtaining a potential solution to a system of equations, it is important to verify that the solution satisfies all original equations. This is typically done by substituting the computed values back into the original matrix equation or system of equations. Verification helps ensure that no computational errors were made during the solution process and confirms that the found solution is valid.

Transcript

00:01 For this problem, we have been given three matrices, a, x, and b.

00:05 First, let's take a look at the matrix equation, a times x equals b.

00:13 From this multiplication, we can get a linear system of two equations into one nodes.

00:19 So let's do this multiplication out and see what we get.

00:25 First we have a, 1, negative 3, 2, 5.

00:31 I'm going to multiply that by x, which is x.

00:33 X sub 1, x sub 2, equaling b, negative 4, and 12.

00:39 Okay, let's do that multiplication out.

00:43 I'm going to be multiplying across the first row, down the first column, and that's going to equal the element in the first row, first column position.

00:54 So that's going to be 1 times x sub 1 plus 2 times x sub 2 equals negative 4.

01:02 Now, onto the second row, first column, and that will give us the second row, first column solution.

01:10 So that's negative 3, x sub 1, plus 5 x sub 2 equals 12.

01:17 And we have a system of equations.

01:20 Now, we could do either substitution or elimination.

01:24 Let's do substitution.

01:26 I'm just going to come over on the side, so i've got some room to work.

01:29 Let's solve that first equation for x sub 1.

01:31 That gives us negative 4 minus 2 x sub 2.

01:36 I'm going to plug that into the second equation.

01:39 So i have negative 3x sub 1 plus 5x sub 2 equals 12.

01:45 And instead of x sub 1, i'm going to substitute in that value that we had for x sub 1...