Inverse of a Matrix - Math is Fun
The inverse of a 2x2 is easy... compared to larger matrices (such as a 3x3, 4x4, etc). For those larger matrices there are three main methods to work out the inverse: Inverse of a Matrix using Elementary Row Operations (Gauss-Jordan) Inverse of a Matrix using Minors, Cofactors and Adjugate; Use a computer (such as the Matrix Calculator) Conclusion
Inverse of a Matrix using Elementary Row Operations (Gauss-Jordan)
The "Elementary Row Operations" are simple things like adding rows, multiplying and swapping ... let's see with an example: Example: find the Inverse of "A": We start with the matrix A, and write it down with an Identity Matrix I next to it: (This is called the "Augmented Matrix")
Inverse of a Matrix using Minors, Cofactors and Adjugate - Math is …
We can calculate the Inverse of a Matrix by: Step 1: calculating the Matrix of Minors, Step 2: then turn that into the Matrix of Cofactors, Step 3: then the Adjugate, and; Step 4: multiply that by 1/Determinant. But it is best explained by working through an …
Matrices - Math is Fun
So we don't divide, instead we multiply by an inverse. And there are special ways to find the Inverse, learn more at Inverse of a Matrix. Transposing. To "transpose" a matrix, swap the rows and columns. We put a "T" in the top right-hand corner to mean transpose:
Solving Systems of Linear Equations Using Matrices - Math is Fun
A is the 3x3 matrix of x, y and z coefficients; X is x, y and z, and; B is 6, −4 and 27; Then (as shown on the Inverse of a Matrix page) the solution is this: X = A-1 B. What does that mean? It means that we can find the X matrix (the values of x, y and z) by multiplying the inverse of the A matrix by the B matrix. So let's go ahead and do that.
Determinant of a Matrix - Math is Fun
The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more. Calculating the Determinant First of all the matrix must be square (i.e. have the same number of rows as columns).
How to Multiply Matrices - Math is Fun
Multiplying a Matrix by Another Matrix. But to multiply a matrix by another matrix we need to do the "dot product" of rows and columns ... what does that mean? Let us see with an example: To work out the answer for the 1st row and 1st column:
Inverse - Math is Fun
Example: the sine function. The sine function sin takes angle θ and gives the ratio opposite hypotenuse . The inverse sine function sin-1 takes the ratio oppositehypotenuse and gives angle θ
Commutative, Associative and Distributive Laws - Math is Fun
Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7. 6×7 + 2×7 + 3×7 + 5×7 + 4×7 = (6+2+3+5+4) × 7 = 20 × 7 = 140
Matrix Rank - Math is Fun
For a square matrix the determinant can help: a non-zero determinant tells us that all rows (or columns) are linearly independent, so it is "full rank" and its rank equals the number of rows. Example: Are these 4d vectors linearly independent?