Matrix Conformability Basic Math for Economics – Refresher 1 Introduction  For larger systems of equations, it’s much more efficient to analyze them using a matrix.  To help with these situations, let’s review some matrix operations.  The first thing to recall about matrix algebra is that matrices must be conformable.  Recall that conformability involves matrices having the appropriate number of rows and columns depending on the desired operation. The conformability requirements differ based on what we do to the matrices. 2 Conformability  Our first conformability rule to consider is when we add two matrices together.  In this case, both matrices must have the same number of rows and columns. Recall that rows go from left to right, whereas columns go from top to bottom. For example, the follow matrix has 2 rows and 3 columns. 1 6 2 4 3 5 3 Conformability  If the matrices are additively conformable, we simply add each element of the matrices together to obtain our new matrix. Consider matrices 𝐴 and 𝐵 below, 1 6 2 2 1 𝐴= 𝐵= 4 3 5 5 6  Adding them together, 1 6 2 2 1 𝐴+𝐵 = + 4 3 5 5 6 3 7 6 = 9 9 8  4 3 4 3 4 Conformability Suppose instead we had matrices 𝐶 and 𝐷 below, 2 6 1 6 2 𝐶= 𝐷= 1 4 4 3 5 5 3  In this case, we cannot add the matrices together because they are not conformable.   Matrix 𝐶 has 2 rows and 3 columns, while matrix 𝐷 has 3 rows and 2 columns. Thus, matrices 𝐶 and 𝐷 are not additively conformable. 5 Conformability  Now let’s talk about multiplying matrices.  First of all, if we multiply a matrix by a scalar (a single number), we simply multiply each element of the matrix by the value of the scalar. (recall that a scalar is just a number outside of a vector – most of the numbers we use are scalars.) After all, you’re just adding the matrix to itself a number of times equal to the scalar.  For example, suppose we multiplied matrix 𝐴 by 3. 1 6 2 3 18 6 3𝐴 = 3 ∗ = 4 3 5 12 9 15 6 Conformability  When a matrix is multiplied by another matrix, it’s important to remember that the order matters.  This is in contrast to when we multiply with scalars.  For instance, multiplying matrix 𝐶 by matrix 𝐷 is not the same thing as multiplying matrix 𝐷 by matrix 𝐶.  In addition, the matrices must be multiplicatively conformable. The number of columns in the first matrix must equal the number of rows in the second matrix. 7 Conformability 1 6 2 2 1 4 𝐴= 𝐵= 4 3 5 5 6 3  Going back to matrices 𝐴 and 𝐵, recall that they were additively conformable.  They are not, however, multiplicatively conformable. If we calculate 𝐴𝐵 (matrix 𝐴 times matrix 𝐵), matrix 𝐴 has 3 columns while matrix 𝐵 has two rows. Thus, they cannot be multiplied. The same result holds if we calculate 𝐵𝐴. Matrix 𝐵 has 3 columns while matrix 𝐴 has two rows. 8 Conformability 2 6 1 6 2 𝐶= 𝐷= 1 4 4 3 5 5 3  While matrices 𝐶 and 𝐷 were not additively conformable, they are multiplicatively conformable.  In fact, they are multiplicatively conformable in both directions. If we calculate 𝐶𝐷, matrix 𝐶 has 3 columns while matrix 𝐷 has 3 rows. Likewise, if we calculate 𝐷𝐶, matrix 𝐷 has 2 columns while matrix 𝐶 has 2 rows. 9 Conformability 1 6 𝐶= 4 3  2 5 2 6 𝐷= 1 4 5 3 Let’s calculate 𝐶𝐷.  We must calculate each element of the matrix individually. To find the first row and first column of matrix 𝐶𝐷, we multiply the elements of the first row of matrix 𝐶 by the elements of the first column of matrix 𝐷, then add them together. That’s a big step. Let’s walk through that one. 10 Conformability 2 6 1 6 2 𝐶= 𝐷= 1 4 4 3 5 5 3  Looking at the first row of matrix 𝐶 and the first column of matrix 𝐷, we can multiply the elements together, then add them up, yielding, 1 ∗ 2 + 6 ∗ 1 + 2 ∗ 5 = 18  Thus, the element in the first row and first column of matrix 𝐶𝐷 is 18. We can repeat this process to find all of the elements of matrix 𝐶𝐷. 11 Conformability 2 6 𝐷= 1 4 5 3 To find the element in the second row and first column of matrix 𝐶𝐷, we use the second row in matrix 𝐶 and the first column in matrix 𝐷.  And so on. Repeat for each possible combination of rows and columns for the two matrices.  This leads us to matrix 𝐶𝐷, 18 36 𝐶𝐷 = 36 51  Note that the resulting matrix will contain the same number of rows as the first matrix and the same number of columns as the second. 1 6 𝐶= 4 3  2 5 12 Conformability  2 6 1 6 2 𝐶= 𝐷= 1 4 4 3 5 5 3 What if instead we calculated 𝐷𝐶?  Now we use the rows from matrix 𝐷 and the columns from matrix 𝐶. This yields, 26 30 34 𝐷𝐶 = 17 18 22 17 39 25 Notice that not only did we obtain a different result from matrix 𝐶𝐷, the matrices do not even have the same dimensions! 13 Conformability  Lastly, we have inversion conformability.  For a matrix to be inversion conformable, it must be square. A square matrix is simply a matrix with the same number of rows and columns.  We’ll go over matrix inversion in a later lesson, however. We need to talk about a few more things before we get there! 14

Matrix Conformability

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