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Transformation notation describes one equation in terms of another.

f(x) = 5x - 2
g(x) = f(x) - 5

g(x) is described in transformation notation because it is described in terms of f(x).

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To apply transformations to matrices, you multiply them by the following matrices:

Reflection across the Y-axis:
[-1         0]
[0          1]

Reflection across the X-axis:
[1         0]
[0        -1]

Rotation 90° counterclockwise about the origin:
[0         -1]
[1          0]

Rotation 180° about the origin:
[-1         0]
[0          -1]

Rotation 270° counterclockwise about the origin:
[0         1]
[-1        0]

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When writing quadrilaterals as a matrix, you take the coordinates of the point and put them in a matrix, where one row or column is a ordered pair and each row or column is an x or y value.

Quadrilateral ABCD
a= (9, 5) b= (10, 2) c= (4, -1) d= (-2, -2)

[9 10 4 -2]
[5 2 -1 -2

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Finding the Additive Inverse of a Matrix

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Finding the Additive Inverse of a Matrix

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The additive inverse of a matrix places the opposite of a number in the slot the number is in.

[1      2]
[3      4]

Additive inverse:

[-1 -2]
[-3 -4]

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Finding the Multiplicative Inverse of a Matrix

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Finding the Multiplicative Inverse of a Matrix

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The Matrix Multiplicative Inverse

The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. In math symbol speak, we have A * A sup -1 = I. This tells you that when you multiply a matrix A with its multiplicative inverse, you will get the identity matrix.

Yes, we write the inverse with a superscript of -1. When we deal with regular numbers, our multiplicative inverse is the number we multiply by to get 1. So, for the number 2, it is 1/2. For the number -3, the multiplicative inverse is 1/3. For our normal numbers, the multiplicative inverse is simply 1 divided by our number.

Unfortunately, not all matrices will have an inverse, nor is finding the multiplicative inverse that simple. In order to find the multiplicative inverse, we have to find the matrix for which, when we multiply it with our matrix, we get the identity matrix. Our matrices must also be square, having the same number of rows and columns.

We will leave the discussion on how to find the inverse of a matrix to another lesson. For this lesson, we will talk about its benefits. You see, it is useful to learn about the multiplicative inverse of a matrix because if we know it, then we can use it to help us solve equations with matrices in them.

A Matrix Equation

For example, we can use it to solve a problem like this:

This matrix equation is in the form of Ax = b, where A is your coefficient matrix, x is your variable matrix, and b is your answer matrix. While we can use other methods to solve such a problem, if we know the multiplicative inverse of our coefficient matrix, then we can easily solve the problem by simply multiplying both sides by the inverse.

So, if we knew A sup -1, our answer would be x = A sup -1 * b. Yes, our answer would be our answer matrix, b, multiplied by the multiplicative inverse of our coefficient matrix. Let's look at how this works. For our coefficient matrix, we have this matrix as the multiplicative inverse matrix:

We can check whether this inverse is real or not by multiplying it with our coefficient matrix to see if we get the identity matrix. Multiplying the two matrices, we see that we do get the identity matrix:

We know for sure now that this inverse is the real inverse, and it works for us.

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multiplicative inverses of matrices and matrix equations - video & lesson transcript | study.com

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Multiplicative Inverses of Matrices and Matrix Equations - Video & Lesson Transcript | Study.com

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The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. In math symbol speak, we have A * A sup -1 = I. This tells you that when you multiply a matrix A with its multiplicative inverse, you will get the identity matrix.

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Solving a System of Equations Using an Inverse Matrix

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Solving a System of Equations Using an Inverse Matrix

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Using Matrix Inverse to Solve a System of 2 Linear Equations

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Using Matrix Inverse to Solve a System of 2 Linear Equations.

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The determinant of a matrix is ad-bc, when:

[a        c]
[b        d]

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A vector that goes up 7 and right 8 is written as:
<8 , 7>
with the x-movement listed first and the y-movement listed second.

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determinant: a useful value that can be determined from a square matrix.

Multiplicative Inverse Matrix: A matrix that when multiplied by the original matrix the product is the multiplicative identity matrix.

Multiplicative Inverse Matrix: A matrix that when multiplied by a matrix results in the same matrix.
[1     0]
[0     1]

Additive Inverse Matrix: A matrix that when added to the original matrix the resulting sum is the additive identity.

Additive Identity Matrix: A matrix that when added to another matrix the resulting sum is the same matrix.
[0    0]
[0    0]

Vector: a quantity that has both magnitude and direction

Magnitude: the distance moved

Direction: the direction in which the object is moved.