In Exercises 37–38, find the products and to determine whether...

Question

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

Accepted Answer

Welcome back. I'm so glad you're here. We've got two matrices A. And B. And we're going to determine whether B. Is the multiplication inverse of A. By finding the products of A. B and B. A. Why do we have to do them both? Well matrix multiplication is not communicative. We don't know for sure that A B will equal B. A. But if B. Is the multiplication inverse of A then they will both equal the multiplication identity matrix. And that's where we've got diagonal one starting in the upper left hand corner and zeros filled in all around it. If they both equal the multiplication of identity in verse then yes we've confirmed that B is the multiplication inverse of A. So let's get started with the multiplication. We're going to take the first row of A. And we are going to multiply that by the first column of B. So that's going to start off from row one of A. We've got one times column wanna be one plus from row one of a zero from column one of B zero plus the last one in row one of a times the last one in column one of B. And now we're going to continue working with AIDS row one So one again from there and now we're looking at bees column two. So one times zero from B's second column plus zero from a. S first row and two from B's second column. And we've got zero from a s first row and we've got one from B's second column continuing with row one from A. We've got one. We're going to work across row one and now we're moving down bees third column. So that's zero plus zero from a. S first row times five from bees third row plus zero from the first row, times three from bees third column. Alright we got one row done and now let's look at a second row. We've got zero from a second row and we're going back down bees first column again. So times one plus three from a second row, times zero from B's first column plus -5 from a second row and zero from B's first column repeating A. Is row two. We've got zero times. Now B's second column zero plus three from a second row and two from B's second column plus negative five from a a second row and one from B's second column, continuing with a second row. We've got zero times. Now we're looking at B's third column, zero plus negative, excuse me. Three again from a second row and going down to five from B's third column plus negative five from a second row and three from B's third column, two of them. Now we'll look at a third row. We've got zero from a third row times one from B's first column negative one from a second row times zero of B's first column and two from a third row times zero of bees first column repeating third row of a. Is zero times. Now we're looking at B. Second column zero plus negative one from a third row, times two from bees second column plus two from a third row, times one from B's second column. One more zero from a. A third row times zero from B's third column negative one from a third row, times five from bees, third column plus two from 83rd row and three from bees. Third column. Nice. Now let's do that. Multiplication, one times one is one plus zero times 00 plus zero times 001 times 00 plus zero times to zero plus zero times one is zero. One time 00 plus zero times five is zero plus zero times three is zero, zero times 10 plus three times 00 plus negative five times zero is zero zero time 00 plus three times two is six plus negative five times one is negative 50 times 00 plus three times five is 15 plus negative five times three is negative 15. Zero times one is zero negative one times zero is 02 times 000 times 00 plus negative one times two is negative, two plus two times one is two. Zero time 00 plus negative one times five is negative five and plus two times three is six. Now let's do the addition one plus zero plus zero is 10 plus zero plus zero is 00 plus zero plus zero is zero, zero plus zero plus zero is 00 plus six minus five is 10 plus 15 minus 15 is 00 plus zero plus 000 plus negative two plus two is 00 plus negative five plus six is one. Hey that checks out. So we know that A times B does equal our multiplication multiplication identity matrix. Now let's work on B times A. I know you're excited to do this again so scooch this down a little bit so we can take a look. I'm gonna make these a little bit smaller And we've got starting with B. And now looking at A. And I'm going to go a little bit faster. So we're starting with these rows and multiplying by eight columns. So we've got one times one plus zero times zero plus zero times zero Repeating B's First Row, one Times zero from a second column, zero times three plus zero times negative one. Continuing with bees first row and now going to a third column, zero times zero plus oops, bees. First row starts with one, zero times negative five plus zero times two. Now going to B's second column, we've got zero times repeating A. S first B's second row times A. Is first column zero times one plus two times zero plus five times zero repeating be second row, zero times zero. Now we're in a second column plus two times three plus five times negative one. And repeating B's 2nd row, zero times zero or on AA 3rd column Plus zero or times negative five plus five times two. And now for bees, third row zero times were a s first column again zero times one plus zero times zero plus This should be a one, three times zero, repeating B's third row, zero times zero from a second column plus one times three plus three times negative one. And repeating B's third row, zero times zero from ace, third column plus one times negative five plus three times two. Alright let's do the multiplication one times one is one plus zero times 00 plus zero times 001 times 00 plus zero times plus zero times negative one is 01 times 00 plus zero times negative 50 plus zero times two is zero, zero times one is zero plus two times 00 plus five times 000 times 00 plus two times three is six plus five times negative one is negative five zero times 00 plus two times negative five is negative. 10 plus five times two is 10. 0 times one is zero plus one times +00 plus three times +000 times +00 plus one times three is three plus three times negative one is negative three. It's always so exciting when you get a real number and then it's like they'll just add up to 00 times 00 plus five times or one times negative five is negative five and three times two is six. Now let's do the addition one plus zero plus zero is 10 plus zero plus 000 plus zero plus 00 zero plus zero plus zero is 00 plus six plus negative five is 10 plus negative 10 plus 10 is +00 plus zero plus +000 plus three plus negative +300 plus negative five plus six is one. Yay, all that to go back up here and say this matches with answer choice A. Which is yes. We did confirm the B is the multiplication inverse of A. Well done. We'll catch you on the next one.

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

Key Concepts

Matrix Multiplication

Multiplicative Inverse

Identity Matrix

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