In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations. {

In Problems 41-60, use the inverses found in Problems 31-40...

Key Concepts

Determinant and Invertibility

For a matrix to have an inverse, its determinant must be non-zero, which indicates that the matrix is invertible. The determinant is a scalar value derived from the matrix components and serves as an indicator of whether a unique solution exists for the system of equations; a zero determinant means the system may have either no solution or infinitely many solutions.

Matrix Representation of Systems

This concept involves expressing a system of linear equations in the form Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the constant column vector. This formulation provides a structured and compact way to handle multiple equations and sets the stage for using various linear algebra techniques to find the solution.

Inverse Matrix Method

This method is used to solve a system of linear equations by multiplying both sides of the matrix equation by the inverse of the coefficient matrix, provided that the inverse exists. The process involves computing the inverse of matrix A so that the solution is given by x = A^(-1)b, which simplifies the problem by reducing it to standard matrix multiplication.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a new matrix. In the context of solving systems of equations, it is used to apply the inverse matrix to the constant vector, thereby yielding the values of the unknown variables. Proper understanding ensures accurate manipulation and solution of matrix equations.

Transcript

0:00 Hello.

00:02 So we consider a to be the matrix containing the coefficients, x to be the matrix containing the variables, and b to be the matrix containing the constants of our given system of equations.

00:17 So therefore we have a is equal to the matrix 6522, x is equal to the column vector x, y, and b is equal to the column vector 7 .2.

00:26 So, the system of equations can then be represented using the matrix equation a -x is equal to b.

00:39 Then we take the inverse of the matrix a and multiply it on both sides of the equation a -x is equal to b.

00:48 So we have a -inverse times a -x is equal to a -inverse times b, meaning a -inverse times a -times -x is equal to a inverse times b, which gives us that the identity times x is equal to a inverse times b.

01:05 So we have that, again, a inverse times a times x is equal to a inverse times b, but a inverse times a is just the identity.

01:22 So we have the two by two identity here times x is equal to, well, a inverse times b.

01:31 A inverse times b.

01:33 Okay.

01:34 So therefore, by the identity property, we have that x is equal to the column vector x, y, is equal to a inverse times b.

01:43 So we find a inverse, which we did already in our last problem, right? we just concatenates our matrix, which.

01:57 The two by two identity get the identity on the left hand side and we see that the inverse here of a or a inverse that we already found was equal to one negative five halves negative one three okay um so then we find while x is equal to um x y which is equal to a inverse times b so therefore we have that x is equal to, well, a inverse times b.

02:38 So we have a inverse, which, again, is 1, negative 5 halves, negative 1, 3.

02:48 And we are multiplying this by b, so times a column vector, 7 .2.

02:56 So we just do the dot product here, right? the i throat, jath column.

03:00 Again, we have a 2.

03:02 This is, again, a 2 by 2 matrix.

03:05 This is a 2 by 1 matrix...

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