Finding the Determinant and Inverse of a 4 X 4 matrix
CHRISTIAN HEADS UP:
1. All math hubs will have something about God in their final paragraph. The final paragraph in this hub addresses what God has given us.
REDUNDANCY IS MINIMIZED:
2. I do my best not to be redundant in these math hubs; therefore, some of the terms and methods used in this hub have been explained in hub 12.1( Solving 4 Simultaneous Equations with a 4 X 5 Matrix).
THE DNA COEFFICIENT MATRIX:
3. We are going to get very familiar with the coefficient matrix in hub 12.1. Not only are we going to calculate its determinant and inverse in this hub but in future hubs we will be finding its eigenvalues, eigenfunction, and eigenvectors. We will then raise our 4 X 4 matrix to the 153rd power. We will also take its square root, 7th root, and raise it to a complex power. What fun! No, really, I mean it. As I said in other hubs, I believe mathematics is God's language, and as we learn His language it will take us to worlds unknown--spectacular worlds that are impossible to see without it.
4 RULES WE WILL FOLLOW:
4. In this hub we will adhere to the following rules: 1. "The row addition operation does not change the value of the determinant." 2. "The row interchange operation changes the sign of the determinant." 3. "The determinant of an upper triangular matrix is the product of its diagonal entries." 4. "If B[ that is matrix B] is obtained from A[ matrix A] by multiplying one row of A by the number c1, then detB = c1detA."
RULE 1 MUST BE CONSIDERED WITH RULE 4--IMPORTANT!
5. I think this is the most important mathematical aspect you will get out of this hub. When I first studied Linear Algebra it took me a long time to figure out what I was doing wrongly. Yes, row addition does not change the determinant AS LONG AS YOU LEAVE THE PLACEMENT ROW ALONE! For example, at L1 of hub 12.1 we did the following operation: 4R3 -- 5R4----->R4, or 4 times row 3 minus 5 times row 4, then put the result onto row 4. That operation is fine when solving for the variables, but it is not fine when solving for the determinant. If we did that operation to find the determinate then it would have changed the determinants value by a factor of --5. Of course we could keep track of those changes but I don't like doing it that way. I leave the placement row alone even when subtracting. So we do it a bit differently as I did at L1 of this hub: --0.8R3 + R4---->R4. Row 4 is my placement row( where I put the result), and I left it alone; i.e. I did not multiply it with anything. Notice I did not use, 0.8R3 -- R4----->R4. The reason why is this is essentially multiplying row 4 by --1, and that would have changed the sign of our determinant. The reason for this modification is we no longer have the molecular mass matrix augmented to the coefficient matrix. In hub 12.1 this gave us a 4 X 5 matrix with 4 equations. It was understood that an equal sign was at each row, and as long as we multiplied the right side of the equal sign with what we did on the left, everything was balanced. In this hub we are working only with the coefficient matrix. In our example row 4 is not multiplied with anything, not even a --1; therefore, rule 4 is not broken. Multiplying row 3 by --0.8 also does not violate rule 4 because after the operation of addition then R3 is put back on R3 as R3, not as --0.8R3; therefore, nothing was changed on row 3.
FINDING THE DETERMINANT AND INVERSE TOGETHER
6. The identity matrix for our 4 X 4 matrix is a 4 X 4 matrix with all elements being zero except for 1's down the main diagonal. Any matrix multiplied by its identity matrix remains the same matrix. This is analogous to any number multiplied by 1 remains the same number. We adjoin the identity matrix to the coefficient matrix as at L1. We move the identity matrix to the left side by using legitimate mathematical operations as those 4 rules listed above. We do this by putting the first 4 columns into reduced row echelon form as explained in hub 12.1. After we have done this, then the last 4 columns of this 4 X 8 matrix will be the inverse matrix. In the process of doing this we will get the first 4 columns in echelon form( also explained in hub 12.1). So at that time we will use Rule 3 to find our determinant. Then we will continue to put the matrix in reduced row echelon form. This will be like getting 2 things done at once. After the explanations above and at hub 12.1, you shouldn't have trouble following the work done in the images below.
7. It really is amazing how easy it is to make a mistake. I make mistakes all the time when doing math but in matrix algebra it seems like it gets into your sub-conscience mind and gets your hand to write the wrong number or wrong sign, and you are oblivious to it. So at L3, L4, L12, L13, L16 and L19 I just meticulously wrote out the row operations to minimize the mistakes.
ECHELON FORM WILL GIVE THE DETERMINANT
8. At L8 we have arrived at the echelon form; therefore, we can use rule 3 to find the determinant: 5 X 1 X 2 X 1 = 10. Remember we switched rows at L8; therefore, rule 2 tells us to change the sign. So the determinant of our 4 X 4 DNA coefficient matrix is --10. We will now continue at L10 to put the matrix into reduced row echelon form. This work is completed at L22.
WE DO NOT NEED TO ADHERE TO RULE 4
9. At L11 notice we did not stick to leaving the placement row alone. The reason for this is we have already found the determinant; therefore, what we do from that time to the end of this problem are legitimate mathematical procedures to move the identity matrix to the left so that we have the inverse matrix on the right.
THE INVERSE MATRIX
10. WOW! You're still with me? Goodie. We moved the identity matrix to the left, which gives us the inverse matrix on the right. If you multiply the inverse matrix by the DNA coefficient matrix you will get the identity matrix. It is analogous to when any number is multiplied with its reciprocal the product( answer) is 1, as for example, 12 X (1/12) = 1. A superb job of explaining how matrices are multiplied was done by t.elia, and a link to her professional and thorough hub is here:
11. The following 3 images with location points LE1 to LE10 are edits I made on 1-14-12 Sat.
WAS R4 LEFT ALONE?
12. Yes R4 was left alone even though it does not look like it. I'll give you an intuitive reason to believe R4 was left alone. At LE1 we have a 2X2 matrix along with a formula to find its determinant, which is --2. If we want to put this matrix in echelon form to find its det. then we multiply R1 by --3 and add the result to R2 getting (0,--2) for the elements for R2. Notice if we multiply the diagonals elements we get, 1 times --2, which is the correct answer for the det. We do this symbolically at LE2 to LE5. Notice we arrive at the correct formula for the det.
--K TIMES R3 CHANGES THE DET.
13. At LE6 we instead use R2 and multiply it by --1/3, add R1 to it, and then place the result on R2. We do the work from LE6 to LE10 and get --K(ad--bc). As you can see we changed the value of our det. by --K, which is in this case a --(1/3). This is why we could not multiply R4 by anything, not even a --1.
GOD'S GIFTS, AND HOW THEY GLORIFY HIM:
14. As mentioned above, mathematics will take us to worlds unknown, but it also GIVES us worlds unknown. The world we live in now is much different than the world our ancestors lived in 150 years ago. Mathematics has had a significant--and in many cases a foundational--role in giving us the world we enjoy now. As God reveals the intricacies of mathematics to us, we can do remarkable things with that knowledge, but more importantly, we are able to understand it. God gave the ostrich one brain( Job 39:13,17), and us another( Job 38:36), and it is these magnificent minds that God created and gave to us that enable us to build the world in which we live and enjoy. Every good thing which we have achieved glorifies God, not us. We can no more take credit for our accomplishments than a computer can take credit for what it can do. The work we have done in this hub( 12.2) and the last one( 12.1) my ti-89 calculator can do in about 1 or 2 seconds. In fact it does everything for me except make the morning coffee, but I know it was the engineers of TI who designed the ti-89, and it is God who designed and made the engineers and gave them their remarkable minds. This entire technological age is a tribute to God. If we as individuals and as a nation continue to sin against God, and then we loose all that we have . . . well, we can take credit for that. The Bible clearly teaches that a nation that turns its back to God, and ignores His commandments, then God will turn against that nation. There will be no more blessings regardless of how smart we are.