# Null, Identity and Inverse Matrices Worksheets

In this section we look at three specific types of matrices. The null or zero matrix is pretty easy to remember because all of the values of all the elements are zero. These can be used to help use define other matrices when we add them to any other array, they do not change the value of any of the elements. Identity matrices have many zeroes as well, they have 1s on their main diagonal and 0s everywhere else. The are found only square patterns. The inverse of a matrix is just like a reciprocal version of it. When you multiply a matrix by its inverse the result will be an identity matrix. These worksheets and lessons introduce students to three unique forms of matrices and how to operate with.

### Aligned Standard: HSN-VM.C.10

- Null Form Step-by-step Lesson- Null means zero, students love that one. Good to start them off with confidence.
- Guided Lesson - An identity matrix, confirming or disavowing inverse matrices, and another null array for you.
- Guided Lesson Explanation - The first and last one goes quick. The second one takes a bit of time.
- Practice Worksheet - Write a null array with fixed orders, make an identity table set, and check for inverses of matrices.
- Matching Worksheet - Yeah, I did kind of take the shotgun approach in putting these together. It came out well though.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Number 1 = Null Matrices, Number 2 = Inverse Matrices, Number 3 = Identity Matrices.

- Homework 1 - Null matrices contain all 0s. In this case, you must write a 3 x 6 matrix. A 3 x 6 table array contains 3 rows and 6 columns.
- Homework 2 - We have inserted the number from our matrix into this format.
- Homework 3 - The diagonal contains all 1s. The order of 4 means that the identity form is a 4 x 4.

### Practice Worksheets

Same topic pattern as the homework, but we provide a little more explanation here.

- Practice 1 - Write an identity table of order 8?
- Practice 2 - Write a null version of order: 4 x 6
- Practice 3 - Check whether an inverse exists for the following matrices.

### Math Skill Quizzes

We made a quiz on each individual skill rather than mixing them up.

- Quiz 1 - While this just seems like you are writing zeroes everywhere, it has a real purpose that we will bring to light in the next topic.
- Quiz 2 - Look at spotting inverses here.
- Quiz 3 - An identity table is a square array where all entries are zeroes except for the top-left to bottom-right diagonal.

### What are Null, Identity, and Inverse Matrices?

A matrix represents a rectangular array of numbers arranged in columns and rows. These numbers are known as the entries or elements of the matrix. The arrangement of elements in rows and columns determine the dimension of a table. The dimensions of any matrix are written as 'm x n', where m represents the number of rows, and n represents the number of columns. We arrange values in this form to help organize them and reduce clutter when we are performing operations or analysis of them. There are several different forms of these arrays that you should investigate because they will give strategies to use when working to solve them.

**Null** - The null matrix is the easiest one to identify. If each entry of an m x n matrix is zero, the table is said to be the zero or null. It is written as 0_{m, n}. [0 0] is a null matrix with 1 x 2 dimensions. These come in many different varieties and can help you better understand the entire system.

**Identity** - A square matrix having all the elements of the main diagonal as 1 with the remaining elements as 0 is known as an identity matrix. Identity variety is also known as a unit matrix. An identity matrix is denoted by I_{n x n}, where n x n represents the dimensions of the matrix.

**Inverse** - The inverse of a matrix A is an array that results in an identity matrix when multiplied by matrix A. The inverse of the array is denoted by A^{-1} This is very comparable to the concept of a reciprocal.