How To Write Augmented Matrix

Introduction to Matrices / Matrix Size (page 1 of 3)

Sections: Augmented & coefficient matrices / Matrix size, Matrix annotation & types, Matrix equality

Augmented matrices

Matrices are incredibly useful things that crop up in many different practical areas. For now, yous'll probably only practise some simple manipulations with matrices, and then you'll move on to the next topic. Merely yous should not be surprised to encounter matrices again in, say, physics or engineering. (The plural "matrices" is pronounced equally "MAY-truh-seez".)

Matrices were initially based on systems of linear equations .

Given the following system of equations, write the associated augmented matrix.

2x + 3y � z = 6
�ten � y � z = nine
ten + y + viz = 0

Write downward the coefficients and the respond values, including all "minus" signs. If in that location is "no" coefficient, and so the coefficient is "1 ".

[ [ 2 3 -1 6 ] [-1 -1 -1 9 ] [ 1 1 6 0 ] ]

That is, given a system of (linear) equations, you can relate to it the matrix (the grid of numbers inside the brackets) which contains merely the coefficients of the linear organization. This is called "an augmented matrix": the grid containing the coefficients from the left-hand side of each equation has been "augmented" with the answers from the correct-mitt side of each equation.

The entries of (that is, the values in) the matrix correspond to the x -, y - and z -values in the original organization, as long equally the original system is arranged properly in the first place. Sometimes, you'll need to rearrange terms or insert zeroes equally place-holders in your matrix.

Given the following system of equations, write the associated augmented matrix.

x + y = 0
y + z = 3
z � x = 2

I beginning need to rearrange the system as:

x + y = 0
y + z = 3
�ten + z = 2

Then I tin write the associated matrix every bit:

[ [ 1 1 0 0 ] [ 0 1 1 3 ] [ -1 0 1 2 ] ]

When forming the augmented matrix, use a zero for any entry where the corresponding spot in the arrangement of linear equations is blank.

Coefficient matrices

If you form the matrix only from the coefficient values, the matrix would expect like this:

[ [ 1 1 0 ] [ 0 1 1 ] [ -1 0 1 ] ]

Higher up, we went from a linear organisation to an augmented matrix. You lot tin can go the other way, too.

Given the following augmented matrix, write the associated linear system.

[ [ 1 3 0 4 ] [ 0 2 -1 5 ] [ 3 0 1 -2 ] ]

Remember that matrices require that the variables be all lined upwards prissy and not bad. And information technology is customary, when you have three variables, to use x , y , and z , in that society. So the associated linear system must be:

10 + iiiy = 4
2y � z = 5
3x + z = �2

The Size of a matrix

Matrices are often referred to by their sizes. The size of a matrix is given in the form of a dimension, much as a room might exist referred to as "a ten-by-twelve room". The dimensions for a matrix are the rows and columns, rather than the width and length. For instance, consider the following matrix A :

[ [ 1 2 3 4 ] [ 2 3 4 5 ] [ 3 4 5 6 ] ]

Since A has three rows and iv columns , the size of A is iii � 4 (pronounced equally "three-by-four").

the rows are the red arrows; the columns are the blue arrows

The rows go side to side; the columns go up and down. "Row" and "column" are technical terms, and are non interchangable. Matrix dimensions are always given with the number of rows showtime, followed by the number of columns. Post-obit this convention, the following matrix B :

...is 2 � three . If the matrix has the same number of rows equally columns, the matrix is said to be a "foursquare" matrix. For instance, the coefficient matrix from in a higher place:

[ [ 1 1 0 ] [ 0 1 1 ] [ -1 0 1 ] ]

...is a iii � iii square matrix.

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Cite this article as:

Stapel, Elizabeth. "Introduction to Matrices / Matrix Size." Purplemath. Available from
https://www.purplemath.com/modules/matrices.htm. Accessed