Solving a System Using an Inverse Matrix Exercises 49-56 ,...

Solving a System Using an Inverse Matrix Exercises $49-56$ , use an inverse matrix to solve (if possible) the system of linear equations.
$$\left\{\begin{aligned}-\frac{1}{4} x+\frac{3}{8} y &=-2 \\ \frac{3}{2} x+\frac{3}{4} y &=-12 \end{aligned}\right.$$

Key Concepts

Determinant and Invertibility

The determinant of a matrix is a scalar value that provides important information about the matrix, including whether it is invertible. A non-zero determinant indicates that the matrix is invertible (has an inverse), while a zero determinant implies that the matrix is singular and does not have an inverse. This check is crucial before attempting to use the inverse matrix method to solve a system of equations.

Matrix Representation of a System

This concept involves rewriting 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 column vector of constants. Representing the system in this concise form allows for the use of matrix operations and methods to solve the equations efficiently.

Inverse Matrix

The inverse matrix of a square matrix A, denoted as A?¹, is a matrix such that when it multiplies A, it yields the identity matrix. For solving systems of equations, if the coefficient matrix A is invertible (non-singular), the solution can be obtained by computing x = A?¹b. This method relies on the existence of the inverse and is a powerful tool in linear algebra.

Transcript

00:01 Over here we've been given a system of linear equations.

00:04 So the equations are given to us as the first one is minus 1 upon 4 x plus 3 upon 8y is equal to minus 2.

00:17 The second equation is 3 upon 2x plus 3 upon 4 y is equal to minus 12.

00:27 Now we just need to solve this system of linear equations in order to obtain the values of x and y over here.

00:35 So to solve this, we'll simply write that in the form of a x equal to b, where matrix a is the coefficient matrix.

00:50 So the coefficients for the first equation are minus 1 upon 4 and 3 upon 8.

00:57 So just write that in this form.

00:59 Similarly, for the second equation, the coefficients are 3 upon 2 and 3 upon 4.

01:06 Now, matrix x will be a column matrix having the variables x and y equal to matrix b.

01:14 B is what b is a column matrix again having the constant terms.

01:19 Or you can see the terms that are written on the arches of each equation.

01:25 Now, see, ax is equal to b or i can say x is equal to.

01:29 A inverse b, isn't it? and we all know the basic formula to find out the inverse of any matrix.

01:36 That is 1 upon ad minus bc which is also known as the determinant of a matrix.

01:44 And over here you just multiplied by the matrix which is obtained by interchanging the elements on the main diagonal.

01:51 And for the remaining two entries you just change the signs...

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