18.06SC Linear Algebra (Fall 2011, MIT OCW): Lecture 21 - Cramer's Rule, Inverse Matrix and Volume

18.06SC Linear Algebra

18.06SC Linear Algebra (Fall 2011, MIT OCW). Taught by Prof. Gilbert Strang, this course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and social sciences, natural sciences, and engineering. This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include: a complete set of lecture videos, summary notes, problem solving videos, and a full set of exams and solutions. (from ocw.mit.edu)

Lecture 21 - Cramer's Rule, Inverse Matrix and Volume

Now we start to use the determinant. Understanding the cofactor formula allows us to show that A^-1 = (1/detA)C^T, where C is the matrix of cofactors of A. Combining this formula with the equation x = A^-1b gives us Cramer's rule for solving Ax = b. Also, the absolute value of the determinant gives the volume of a box.

References

Cramer's Rule, Inverse Matrix and Volume
Now we start to use the determinant. Lecture Video and Summary. Suggested Reading. Problem Solving Video.

Go to the Course Home or watch other lectures:

I. Ax=b and the Four Subspaces

Lecture 01 - The Geometry of Linear Equations

Lecture 02 - An Overview of Key Ideas

Lecture 03 - Elimination with Matrices

Lecture 04 - Multiplication and Inverse Matrices

Lecture 05 - Factorization into A = LU

Lecture 06 - Transposes, Permutations, Vector Spaces

Lecture 07 - Column Space and Nullspace

Lecture 08 - Solving Ax = 0: Pivot Variables, Special Solutions

Lecture 09 - Solving Ax = b: Row Reduced form R

Lecture 10 - Independence, Basis and Dimension