veriify caley Hamilton theorum for the matrix
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Definition of the Cayley-Hamilton Theorem
Brother and sister, Matt and Poly, have been studying math. Matt enjoys matrices while Poly likes polynomials. The siblings are surprised to learn about polynomials of matrices. Specifically, the Cayley-Hamilton theorem shows how a special polynomial of a matrix is always equal to 0. In this lesson, we define and give examples of this theorem.
How the Theorem Works
Matt and Poly have encountered the square matrix, identity matrix, determinant, and characteristic polynomial. If any of this is unfamiliar, don't worry. The examples will help.
An identity matrix is a square matrix with 1s along the main diagonal and 0s everywhere else. A square matrix has an equal number of rows and columns. This lesson deals exclusively with square matrices (as does the Cayley-Hamilton theorem). Here is a 2-by-2 (2 rows and 2 columns) identity matrix:

Here's a 3-by-3 identity matrix:

How about multiplying the identity matrix by a number? To keep things general, the number is the variable λ. So λI for a 2-by-2 case is:

λ multiplies each entry of the matrix: λ times 1 is λ; λ times 0 is 0.
For now, use letters to define a 2-by-2 matrix A:

There are just two more steps for the characteristic polynomial. First, Matt calculates A - λI:

Subtracting one matrix from another involves subtracting the terms at the same locations. The first row, first column of A minus the first row, first column of λI results in a-λ in the first row, first column of the result. Same idea applies for the rest of the subtraction.
Last step is calculating the determinant. For the matrix A - λI, the determinant is (a - λ)(d - λ) - cb. There are more general ways to calculate the determinant for larger matrices but for a 2-by-2 matrix, the product of the terms along the main diagonal minus the product of the terms along the other diagonal is the recipe.

Poly expands and collects terms. The (a - λ)(d - λ) - cb becomes λ2 - (a + d) λ + ad - cb. This result is called the characteristic polynomial and is labeled p(λ) and is the determinant of the A-λ matrix where the identity matrix I has 1s along the main diagonal and 0s everywhere else.
Brother and sister, Matt and Poly, have been studying math. Matt enjoys matrices while Poly likes polynomials. The siblings are surprised to learn about polynomials of matrices. Specifically, the Cayley-Hamilton theorem shows how a special polynomial of a matrix is always equal to 0. In this lesson, we define and give examples of this theorem.
How the Theorem Works
Matt and Poly have encountered the square matrix, identity matrix, determinant, and characteristic polynomial. If any of this is unfamiliar, don't worry. The examples will help.
An identity matrix is a square matrix with 1s along the main diagonal and 0s everywhere else. A square matrix has an equal number of rows and columns. This lesson deals exclusively with square matrices (as does the Cayley-Hamilton theorem). Here is a 2-by-2 (2 rows and 2 columns) identity matrix:

Here's a 3-by-3 identity matrix:

How about multiplying the identity matrix by a number? To keep things general, the number is the variable λ. So λI for a 2-by-2 case is:

λ multiplies each entry of the matrix: λ times 1 is λ; λ times 0 is 0.
For now, use letters to define a 2-by-2 matrix A:

There are just two more steps for the characteristic polynomial. First, Matt calculates A - λI:

Subtracting one matrix from another involves subtracting the terms at the same locations. The first row, first column of A minus the first row, first column of λI results in a-λ in the first row, first column of the result. Same idea applies for the rest of the subtraction.
Last step is calculating the determinant. For the matrix A - λI, the determinant is (a - λ)(d - λ) - cb. There are more general ways to calculate the determinant for larger matrices but for a 2-by-2 matrix, the product of the terms along the main diagonal minus the product of the terms along the other diagonal is the recipe.

Poly expands and collects terms. The (a - λ)(d - λ) - cb becomes λ2 - (a + d) λ + ad - cb. This result is called the characteristic polynomial and is labeled p(λ) and is the determinant of the A-λ matrix where the identity matrix I has 1s along the main diagonal and 0s everywhere else.
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