2.3: The ijth Entry of a Product
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In previous sections, we used the entries of a matrix to describe the action of matrix addition and scalar multiplication. We can also study matrix multiplication using the entries of matrices.
What is the ijth entry of AB? It is the entry in the ith row and the jth column of the product AB.
Now if A is m×n and B is n×p, then we know that the product AB has the form [a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn][b11b12⋯b1j⋯b1pb21b22⋯b2j⋯b2p⋮⋮⋮⋮bn1bn2⋯bnj⋯bnp]
The jth column of AB is of the form [a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn][b1jb2j⋮bnj]
Therefore, the ijth entry is the entry in row i of this vector. This is computed by ai1b1j+ai2b2j+⋯+ainbnj=n∑k=1aikbkj
The following is the formal definition for the ijth entry of a product of matrices.
Let A=[aij] be an m×n matrix and let B=[bij] be an n×p matrix. Then AB is an m×p matrix and the (i,j)-entry of AB is defined as (AB)ij=n∑k=1aikbkj
In other words, to find the (i,j)-entry of the product AB, or (AB)ij, you multiply the ith row of A, on the left by the jth column of B. To express AB in terms of its entries, we write AB=[(AB)ij].
Consider the following example.
Compute AB if possible. If it is, find the (3,2)-entry of AB using Definition 2.3.1. A=[123126],B=[231762]
Solution
First check if the product is possible. It is of the form (3×2)(2×3) and since the inside numbers match, it is possible to do the multiplication. The result should be a 3×3 matrix. We can first compute AB: [[123126][27],[123126][36],[123126][12]]
Now using Definition 2.3.1, we can find that the (3,2)-entry equals 2∑k=1a3kbk2=a31b12+a32b22=2×3+6×6=42
You may wish to use this method to verify that the rest of the entries in AB are correct.
Here is another example.
Determine if the product AB is defined. If it is, find the (2,1)-entry of the product. A=[231762000],B=[123126]
Solution
This product is of the form (3×3)(3×2). The middle numbers match so the matrices are conformable and it is possible to compute the product.
We want to find the (2,1)-entry of AB, that is, the entry in the second row and first column of the product. We will use Definition 2.3.1, which states (AB)ij=n∑k=1aikbkj
Hence, (AB)21=29.
You should take a moment to find a few other entries of AB. You can multiply the matrices to check that your answers are correct. The product AB is given by AB=[1313293200]