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

The determinant of a matrix is closely linked to its invertibility: a square matrix has an inverse if and only if its determinant is nonzero. This concept is critical as it not only determines whether a unique solution exists for the system of equations but also informs the applicability of the inverse matrix method.

Solving Systems using Matrix Inversion

When the coefficient matrix A of a linear system is invertible, the unique solution for the unknown vector x can be computed directly using the formula x = A?¹b. This method leverages the properties of matrix multiplication and inverses to simplify and solve linear systems, making it an important tool in linear algebra.

Matrix Inverse

A matrix inverse is a fundamental concept in linear algebra where, for a given square matrix A, the inverse A?¹ is defined such that A · A?¹ equals the identity matrix. This concept allows us to ‘undo’ the effect of a matrix transformation and is crucial for solving linear systems when the coefficient matrix is invertible.

Matrix Representation of Linear Systems

Representing a system of linear equations in matrix form involves writing the system as Ax = b, where A is the coefficient matrix, x is the column vector of unknowns, and b is the column vector of constants. This formulation facilitates the use of algebraic methods, like the inversion of A, to solve the system efficiently.

Transcript

00:01 Alright, we're going to use the inverse to solve the following problem using this formula on the right.

00:06 So the matrix a times the matrix x equals the matrix b.

00:09 And if we want to find what x equals, then we just use the inverse of a times b.

00:15 So what that means is our matrix a is our equations.

00:25 Our matrix x is our variables.

00:30 Our matrix b is our answers and our inverse are from our previous section here.

00:39 So we're looking for the inverse of 2 .1aa.

00:42 I've got them all here on my second page.

00:45 So 2 .1a, our inverse was 1, negative 1 over a, negative 1 over a, and 2 over a.

00:52 So we're going to use 1, negative 1 over a, negative 1 over a, and 2 over a.

00:59 And 2 over a.

01:01 So all we need to do is we're going to multiply that inverse by our answers.

01:13 Negative 3 and a.

01:18 So that gives us 1 times negative 3 plus negative 1 over a times a and negative 1 over a times negative 3 plus 2.

01:34 Over a times a.

01:39 So 1 times negative 3 is negative 3.

01:43 Negative 1 over a times a is negative 1.

01:46 So negative 3 plus, oh sorry this is a negative a...

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