Algebra 2 Matrix Operations

Welcome to Educator.com.0000

In today's lesson, we are going to continue on with matrices, and this time doing operations on matrices.0002

Just as you can perform mathematical operations on numbers (such as addition, subtraction, and multiplication), you can do the same thing with matrices.0009

Starting out with addition of matrices: addition is defined only for matrices with the same dimensions.0018

So, just to review: when we talk about dimensions, the dimensions of a matrix are m times n,0026

where m is the number of rows and n is the number of columns.0033

So, two matrices must have the same number of rows and columns in order for addition to occur.0040

If that specification is met, then you add the corresponding elements of the two matrices.0049

The sum matrix will have the same dimensions as the original matrices.0055

To illustrate this, let's say that I have two matrices, A, and then I have a matrix B; and I want to add those together.0062

Well, let's look at my first matrix, A; and it will be 4, 0, 3, 1, -2, -4; OK.0074

And I have a second matrix, B, that I am going to add (A + B).0085

OK, so first looking at the dimensions to make sure that I am allowed to even add these: there are 2 rows on this one and 1, 2, 3 columns.0101

Therefore, this is a 2x3 matrix; matrix B has 2 rows and 1, 2, 3 columns; this is a 2x3 matrix.0109

Since these have the same dimensions, they can be added.0119

The sum matrix (the resulting matrix) is also going to have the dimensions of 2x3: it has the same dimensions as these two original matrices.0123

Now, to add these, I am going to add corresponding elements.0133

Recall that corresponding elements occupy the same position in the matrix.0136

So, if I am looking at this position, this is 1, 2; so it is row 2, and in columns, 1, 2; so it is row 2, column 2.0140

The corresponding element over here is also going to be in this position of row 2, column 2.0149

So, all I am going to do is add corresponding elements: 4 + 5 right here, and then I am going to have 0 and -1;0155

over here, 3 + 2; and down here, 1 + 0; in this position, -2 + 3, and in this position, -4 + -5.0169

OK, if I work out the addition on that, that is going to give me 9, -1, 5, 1, (-2 + 3) is 1, (-4 + -5) is -9.0185

So again, first always verify that the two matrices you are being asked to add have the same dimensions.0202

If they do, then you just add the corresponding element and put it in that same position0208

to get the sum matrix, which will have the same dimensions as the two originals.0216

Matrix subtraction is very similar: again, matrix subtraction is defined only for matrices with the same dimensions.0222

To subtract two matrices, subtract the corresponding elements of the two matrices.0230

The result--the difference matrix--will have the same dimensions as the original matrix.0237

So, it's the same concept as with matrix addition.0241

So, looking at that situation here: if I have 3, 2, 4, -1, 0, 6, -2, 0, and this is A;0244

and I have a second matrix that I am going to call B, and I am asked to find A - B;0267

OK, this is the difference matrix, which will be A - B, over here.0276

OK, so first verify that these have the same dimensions: I have 1, 2, 3, 4 rows, and I have 2 columns.0283

Over here, I have 1, 2, 3, 4 rows and 2 columns.0292

My difference matrix is also going to have 4 rows and 2 columns.0299

OK, so I simply subtract: 3 - 6 is going to give me -3; the corresponding elements 2 - 0 is 2; 4 - -5...0303

if I have 4 minus -5, that is going to give me 9; -1 - 6 is going to give me -7; 0 - 2 is going to give me -2;0317

6 - -1 is going to give me 7 (a negative and a negative are going to give me a positive); -2 - 0 is -2, and 0 - 1 is -1.0341

So again, verify that the two matrices have the same dimensions, and then simply0354

subtract corresponding elements; and since we are doing subtraction, you need to be very careful with the signs.0360

OK, now we are going to talk about scalar multiplication.0368

And scalar multiplication is not the multiplication of one matrix times another.0370

Matrix multiplication, we will actually cover in a separate lecture.0376

This is called scalar multiplication, because what you are going to be doing is multiplying a matrix by a constant.0380

And that constant is referred to as a scalar.0387

So, for example, if I am given a matrix that looks like this, and I am asked to multiply it by 2;0391

well, 2 is a constant called a scalar; and let's say this matrix is called A.0402

If I am asked to find 2A, then I am going to multiply the scalar by the matrix.0408

In order to do that, you multiply each element of the matrix by the scalar.0414

So, each element of the matrix, you multiply by the scalar to get the result.0419

OK, therefore, I am going to say 2 times 2 for this position; for this second position in row 1, column 2, it is 2 times 3.0425

Here, it is 2 times -1; 2 times 0; 2 times 4; and finally, 2 times 1.0437

Doing the multiplication out will give me 4; 2 times 3 is 6; 2 times -1 is -2; 2 times 0 is 0; 2 times 4 is 8; and 2 times 1 is 2.0451

And as you can see up here, the scalar product matrix has the same dimensions as the original.0468

This original matrix had 1, 2, 3 rows and 2 columns.0473

The scalar product over here, 2A, has 1, 2, 3 rows and 2 columns--the same dimensions as the original.0480

Again, for scalar multiplication, simply take the scalar (the constant) and multiply it by each element in the original matrix.0493

Just as we have certain properties when we are working with operations on regular numbers and variables,0503

there are also properties that regulate matrix operations.0510

So, if A, B, and C are matrices with the same dimensions, and k is a scalar, then the following hold.0514

This first one, you will recognize as the commutative property of addition.0522

Recall that with numbers, we have the commutative property, where, if you want to add 3 + 2,0530

you can add it in the other order--2 + 3--and get the same result.0535

And the same is true if you are adding two matrices.0539

So, matrix addition follows the commutative property.0542

The next property you may recognize as the associative property for addition of matrices.0548

What this is stating is that I can do these operations in either order.0562

I can either add the two matrices A + B together, then add C to those; or I can add B + C together first, and then add A to those two.0567

So, I can perform these operations in either order; and the result will be the same.0578

Now, looking at scalar multiplication, combined with the addition of two matrices here:0584

this follows the distributive property, and what this is stating is that I can add matrix A and matrix B,0591

and then multiply this scalar by that result; or I can multiply the scalar times one matrix, multiply the scalar by the other matrix,0604

and then add those two together; and I will get the same result.0615

So, this is the distributive property, which you have seen previously.0618

First example: we are going to add two matrices.0625

First, verify that addition is allowed: do these have the same dimensions? two rows, three columns--OK, so far, so good.0628

1, 2 rows; 3 columns: since these have the same dimensions, then addition is allowed.0637

OK, so I am rewriting these down here so we can work with them more easily.0646

Recall that, for addition, you are going to add corresponding elements; you are going to add the elements occupying the same row and column number.0657

So, 2 + -4 is going to give me -2; -1 + 6 is going to give me 5; 3 + -2 is going to give me 1.0671

0 and -3 is going to give me -3; 6 and 0--I am going to get 6; and then, 4 + 4 is going to give me 8.0692

So, all I have to do to find the sum matrix is to add the corresponding elements of these two matrices to get the result.0705

Example 2 involves subtraction: first, verify the dimensions--1, 2, 3 rows, 1, 2, 3 columns.0718

So, this is a matrix with three rows and three columns, so it is a square matrix, since it has the same dimensions in both directions.0727

Here, I have 1, 2, 3 rows and 1, 2, 3 columns; so this is also a 3x3 matrix; so it is a square matrix, since it has the same dimensions in both directions.0740

Therefore, I can subtract these; when I go ahead and subtract, I am going to get a difference matrix that is also 3x3.0755

Beginning with these two corresponding elements: 2 - -1 (I have to be very careful, when I am working with negatives,0772

that this becomes 2 + 1, which is 3, so I get 3 right here); -1 - -2 is going to give me -1 + 2, which is 1.0779

Here, I just have 6 - 3; that is 3; 3 - 0--that is 3; 2 - 4--and that is just going to be...2 - 4 is going to give me -2.0801

0 - -8 (working down here) is going to be 0 + 8, which equals 8 in this position.0818

1 - 6 is -5; 4 - 0 is 4; and then, 3 - -1 gives me 3 + 1, which equals 4.0831

So again, with subtraction, just be really careful with the signs.0845

And I end up with a difference matrix right here that resulted from taking an element0848

in the first matrix and subtracting the corresponding element of the other matrix from that.0856

Example 3: Find the product--and it is the product of a scalar and a matrix, so this is scalar multiplication.0864

And here, the scalar is -2.0873

I am rewriting this down here.0876

Recall that, in scalar multiplication, you are going to multiply the scalar times each element in the matrix,0884

and the result is going to be a scalar product that has the same dimensions as the original.0890

My original here is a 3x3 matrix; so I am going to take -2 times 2 for the first position; -2 times -1;0895

-2 times 6; -2 times 3; and continue on, multiplying each one...-2 times 0; the third row:0909

-2 times 1; -2 times 4; and finally, -2 times 3.0924

Then, when I do my multiplication, I am going to end up with -4, 2, -12, -6, here is -4, 0, -2, -8, and then -6.0932

So again, in scalar multiplication, simply multiply each element in the matrix by the scalar0956

to get a scalar product matrix with the same dimensions as the original.0963

Example 4 is slightly more complicated: we are asked to find -3A + 6B.0970

OK, so what I need to do is multiply matrix A by -3; multiply matrix B by 6; and then add those together.0978

Just to make sure I can add them eventually, I verify the dimensions as 3 rows, 2 columns and 3 rows, 2 columns.0989

So, I will be able to add them.1000

OK, just to note, looking at the distributive property, it says that k, a scalar, times A + B (these two matrices) equals KA + KB.1002

So, there is actually another way I could do this: I could say, "All right, I am going to factor out the -3, and then I am going to end up with A - 2B."1015

This would be another way to do that; however, it is debatable which way is easier.1038

I am going to go ahead and just follow this original.1044

But you actually, if you felt like this way was easier, could have done it this way.1046

OK, so starting out, the first thing we are going to need to do is multiply the A by the scalar -3.1053

OK, and from that I can find 3A; I want to find 3A.1082

So, this is -3 and A, and I want to find -3A.1089

OK, so recall that all we are going to do is multiply each element of the matrix by the scalar.1097

And that will give me 3 here; -3 times 0 is 0; -3 times 2 is -6; -3 times -1 is 3; -3 times 3 is 9.1105

And let's see, then I have -3 times 4, which is -12; OK.1122

Now, what I also need to do is find 6B; so here, I have 3A--I need to find 6B.1128

So, I have B over here; 0, -3, -4, 6, 4, and 9; so, I have B, and I am going to multiply it by 6, and that is going to give me 6B.1139

So, figuring this out, it is: 6 times 0 is 0; 6 times -3 is -18; 6 times -4 is -24.1157

6 times 6 is 36; 6 times 4 is 24; and 6 times 9--that is 54.1171

All right, so here I have 3A; here I have 6B; now, I need to add those.1178

So, let me go ahead and copy 6B right over here, so I can add it.1184

OK, 0, -18, -24, 36, 24, and 54: now, I am erasing that; these two are no longer equal.1200

OK, all we need to do with matrix addition is to add the corresponding elements.1215

So, I am going to add 3 and 0; and this is going to give me 3.1222

I am going to add 0 and -18 to get -18; -6 and -24 is -30; 3 and 36 is 39;1230

9 and 24...or excuse me, this actually should be -9; correct that--that is -3 times 3; that is -9 + 24 is 15;1241

and then, I have -12 and 54, to give me 42.1251

So, this here is -3A + 6B: so right here is my solution.1256

And what I did is took A; I multiplied it by the scalar -3 to get this -3A.1264

I took 6, and I multiplied it by the second matrix, B, to get 6B.1272

Then, I added -3A + 6B to get -3A + 6B as my solution.1279

That concludes this lesson on matrix operations at Educator.com; I will see you next lesson.1291