Matrices and Systems of Linear Equations

Matrices and Systems of Linear Equations
Matrices and Systems of Linear Equations
The rows of A are its horizontal lines of entries, so A has three rows. The
columns of A are its vertical lines of entries, so A has two columns. The size
(also called order) is 3 × 2, its number of rows by its number of columns.
314. Recall that the element aij
is in the ith row and jth column of matrix A.
(A) a13 = 4
(B) a31 = –4
(C) a35 = –7
(D) a14 = 3
(E) a33 = 8
315. (A) A column vector is a matrix with only one column, Thus, the 3 × 1
column vector whose elements are 2, 3, and 8 is .
(B) A row vector is a matrix with only one row. Thus, the 1 × 4 row
vector whose elements are 0,–2, 1, 5 is [0 –2 1 5].
(C) For any square matrix A = [aij]n×n
, the main diagonal elements are
, a22
, a33
, …, ann
Matrices and Systems of Linear Equations
. Note: Hereafter a matrix’s main diagonal will be
called simply its diagonal. Thus, the 3 × 3 matrix A = [aij]3×3 whose
diagonal elements are 3,–4, 5 and whose off-diagonal elements are
1’s is .
(D) I3×3 denotes the 3 × 3 identity matrix. An identity matrix is a square
matrix whose diagonal elements are 1’s and whose off-diagonal
elements are 0’s. Thus, I3×3 = .
(E) All the elements of the m × n zero matrix, denoted 0 (or 0m×n
), are
zero. Thus, the 2 × 3 zero matrix is .
316. Matrices can be added (subtracted) only if they are the same size. The sum
(difference) of two matrices is the matrix whose elements are the sums
(differences) of the corresponding elements of the two matrices.
317. Refer to question 316.
318. For a matrix A = [aij]m×n and a scalar k, kA = [kaij]m×n
is the product A by
k. Thus,
319. The product of a 1× n row vector and a n ×1 column vector is obtained by
multiplying corresponding elements and adding the resulting products. Thus,
Note: The product is undefined if the row and column vectors do not have the
same number of elements.
320. If A = [aij ]m×k
is a matrix of size m × k and B = [bij ]
is a matrix of size
k × n so that the number of columns of A is equal to the number of rows of B,
then the product AB is the m × n matrix C = [cij ]m×n whose ijth element is the
sum of the products of the corresponding elements of the ith row of A and the jth
column of B. When this inner matching of sizes occurs, the matrices A and B are
compatible for multiplication. Thus,
Note: The product AB is not defined if the number of rows of A is not equal to
the number of columns of B; and, in general, AB ≠ BA.
321. Proceed as in question 320.
322. Proceed as in question 321.
(C) No.
323. The determinant of a square matrix A, denoted det A or |A|, is a scalar
associated with the matrix and its elements. Note: The determinant is not
meaningful for non-square matrices.
A second order (2 × 2) determinant is defined as follows: det = ad –
Matrices and Systems of Linear Equations
Thus, = (–4)(2) – (5)(1) = –8 – 5 = –13.
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