Let's prove the theorem (MN)T = NT MT. Note: the following is a common technique for...

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Let's prove the theorem (MN)T = NT MT. Note: the following is a common technique for proving matrix identities. (a) Let M = (m) and let N = (n). Write out a few of the entries of each matrix in the form given at the beginning of section 7.3. (b) Multiply out MN and write out a few of its entries in the same form as in part (a). In terms of the entries of M and the entries of N, what is the entry in row i and column j of MN? (c) Take the transpose (MN) and write out a few of its entries in the same form as in part (a). In terms of the entries of M and the entries of N, what is the entry in row i and column j of (MN)T? (d) Take the transposes NT and MT and write out a few of their entries in the same form as in part (a). (e) Multiply out NT MT and write out a few of its entries in the same form as in part a. In terms of the entries of M and the entries of N, what is the entry in row i and column j of NTMT? (f) Show that the answers you got in parts (c) and (e) are the same. Let's prove the theorem (MN)T = NT MT. Note: the following is a common technique for proving matrix identities. (a) Let M = (m) and let N = (n). Write out a few of the entries of each matrix in the form given at the beginning of section 7.3. (b) Multiply out MN and write out a few of its entries in the same form as in part (a). In terms of the entries of M and the entries of N, what is the entry in row i and column j of MN? (c) Take the transpose (MN) and write out a few of its entries in the same form as in part (a). In terms of the entries of M and the entries of N, what is the entry in row i and column j of (MN)T? (d) Take the transposes NT and MT and write out a few of their entries in the same form as in part (a). (e) Multiply out NT MT and write out a few of its entries in the same form as in part a. In terms of the entries of M and the entries of N, what is the entry in row i and column j of NTMT? (f) Show that the answers you got in parts (c) and (e) are the same.