Matrix Concepts and Types

Questions and Answers

Define what a matrix is.

A matrix is a rectangular array of elements.

Identify special types of matrices.

Row Vector, Diagonal Matrix, Column Vector, Identity Matrix, Submatrix, Zero Matrix, Square Matrix, Tri-diagonal, Upper Triangular Matrix, Lower Triangular Dominant Matrix.

Identify when two matrices are equal.

Two matrices are equal when they have the same dimensions and corresponding elements are equal.

What is a vector?

A vector is a matrix that has only one row or one column. Signup and view all the answers

Which of the following is an example of a row vector?

[25 20 3 2 0] Signup and view all the answers

Identify an upper triangular matrix:

[10 -7 0; 0 -0.001 6; 0 0 15005] Signup and view all the answers

Give an example of a square matrix.

[25 20 3; 5 10 15; 6 15 7] Signup and view all the answers

What is the size of matrix [A] with 3 rows and 4 columns?

3×4 Signup and view all the answers

Study Notes

Definition of a Matrix

A matrix is a rectangular array of elements, which can include symbolic expressions or numbers.
Denoted mathematically as [ A], each entry is referred to as an element and is represented as aij, where i is the row number and j is the column number.

Types of Matrices

Row Vector: A matrix with one row, denoted as [ B] = [b1 b2 ... bn].
Column Vector: A matrix with one column, denoted as
```
[C] = 
[ c1 ]
[ c2 ]
[ ... ]
[ cm ]
```
Square Matrix: A matrix where the number of rows m is equal to the number of columns n. Diagonal elements are referred to as a11, a22, ..., ann.
Zero Matrix: A matrix where all elements are zero.
Diagonal Matrix: A square matrix where all off-diagonal elements are zero.
Identity Matrix: A diagonal matrix where all diagonal elements are 1.
Upper Triangular Matrix: A matrix where all elements below the main diagonal are zero.
Lower Triangular Matrix: A matrix where all elements above the main diagonal are zero.
Tri-diagonal Matrix: A matrix that only has non-zero elements on the main diagonal and the diagonals directly above and below it.
Dominant Matrix: A matrix where each diagonal element is greater than the sum of the absolute values of the other elements in its row.

Matrix Size

The size of a matrix [A] is denoted by m×n, indicating m rows and n columns. For a matrix with 3 rows and 4 columns, the size is 3×4.

Matrix Equality

Two matrices are equal if they have the same dimensions and corresponding elements are equal.

Submatrices

A submatrix is formed by deleting some rows and/or columns from a matrix, without removing any rows or columns completely.

Example Matrices

Illustration of tire sales data represented in matrix form showing sales by tire brands over four quarters.
Example matrices include a row vector [ B] = [25 20 3 2 0] and a column vector
```
[C] = 
[25]
[5]
[6]
```

Additional Key Points

Identifying sales in a matrix, such as finding the number of Copper tires sold in Quarter 4 by referencing the corresponding row and column.
Key features of specific matrix types like identifying upper triangular matrices through the positioning of zeros below the diagonal.

Description

This quiz covers the fundamentals of matrices, including their definitions and special types. You will also learn how to determine when two matrices are equal. Understanding matrices is essential for various applications in mathematics and data presentation.