Solve the matrix equation for X [ 1 -1 1 2 3 0 0 2 -1 ]

Solve the matrix equation for X [ 1 -1 1 2...

Key Concepts

Matrix Equations

A matrix equation is an equation that involves matrices and operations between them, typically taking the form AX = B, where A and B are known matrices and X is the matrix to be determined. This formulation provides a compact way to represent systems of linear equations and allows for the use of matrix operations in finding solutions.

Inverse Matrix

An inverse matrix, when it exists, is a matrix that can be multiplied with the original matrix to yield the identity matrix. For a square matrix A, if there exists a matrix A?¹ such that A?¹A = AA?¹ = I, then A is said to be invertible. This property is crucial in solving matrix equations because it allows for the expression X = A?¹B.

Matrix Multiplication

Matrix multiplication is the process of combining two matrices to form a new matrix, where the entries of the product are determined by the dot products of rows and columns from the two matrices. It is a fundamental operation in linear algebra that is essential for expressing and solving matrix equations, including the operation of applying an inverse matrix.

Systems of Linear Equations

Systems of linear equations can be represented in matrix form as AX = B, where the matrix A contains the coefficients, X is the vector or matrix of unknowns, and B is the vector or matrix of constants. This representation facilitates the application of various algebraic techniques, such as substitution with the inverse matrix or elimination methods, to find the solution to the system.

Transcript

00:01 For this problem, we have been given a matrix equation, a 3x3 matrix times x, equaling a 3 by 5 matrix.

00:08 And our goal is to solve this for x.

00:11 Now, if you go back and look at theorem 1 .6 .2, it says that if i have an equation with matrix multiplication, matrix a times x equaling matrix b, i can solve this for x by taking the inverse of matrix a times matrix b.

00:29 So that's what we have here.

00:31 The 3 by 3 matrix is a, and i'm multiplying it by matrix x, and b is my 3 by 5 matrix.

00:39 So let's see if we can find the inverse of matrix a, because then it's just matrix multiplication to find our answer.

00:46 So in order to find our inverse matrix, first we just copy down matrix.

00:55 And i'm going to augment this with the identity matrix.

00:59 In order to find the inverse, the goal is to use row manipulation to make the left -hand side look like the identity matrix.

01:10 Once that happens, the right -hand side will have transformed into the inverse of matrix a.

01:16 So that's my goal, to make the left -hand side look like the identity matrix.

01:22 So let's begin.

01:25 Now, if you recall, the identity matrix has a string of ones in a diagonal from the top left to the bottom right.

01:32 Every other element is going to be zero.

01:35 So if i look at the first column, i need to have an entry in the first row position.

01:41 So i'm going to leave the first row untouched.

01:46 Everything else in the first column, i would like to set equal to zero.

01:50 Now, if you look at the third row, it already has a zero there.

01:54 So i don't have to do any manipulation.

01:57 I can just copy that row just as is.

02:00 I do, however, need to do something to the second row.

02:03 I have to get rid of that 2.

02:05 So what i can do here is i will take, sorry, i'll take negative 2 times the first row, and i'll add that to the second row.

02:16 They'll give me negative 2 plus 2, that will allow me to put a 0 in that spot.

02:21 If i do that manipulation all the way across my row, i get 0, 5, negative 2, negative 2, 1, and 0.

02:32 First column is done.

02:34 Now let's look at the second column.

02:37 The second column needs to have an entry in the second row.

02:42 So i'm going to leave the second row untouched.

02:48 Everything else in the second column, i want equal to zero.

02:52 So to get rid of the negative one in the first row, i can multiply that by five times the first row, and i'll add it to the second row.

03:07 And to get rid of the two in my last row, i can take negative 2 times the second row plus 5 times the third row.

03:16 That'll give me a negative 10 plus 10.

03:19 That's going to equal 0.

03:21 Doing that all the way across both rows gives me a new matrix.

03:25 I'll do this in black.

03:26 I have 503 -3 -310.

03:32 And the bottom row will be 0 -0 -0 -9 -1, 4, negative 2, and 5.

03:39 Okay, first two columns are done.

03:41 We're going to look at the third column.

03:44 Now the third column needs to have an entry in the third row...

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