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Math 141 Week in Review
Week 3 Key Topics
Section 2.4
Matrix Size and Entries
• A matrix is a rectangular array of numbers enclosed in brackets.
• The number of rows and columns gives the size of a matrix.
o A matrix with 2 rows and 3 columns is a 2×3 matrix.
o In general, a matrix with n rows and m columns is an n×m matrix.
• Specific entries in the matrix are denoted using subscripts.
o The entry in row 3 and column 1 of a matrix A is called a31 .
o In general, an entry in row i and column j of a matrix A is called aij .
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A square matrix has the same number of rows and columns.
Two matrices are equal if they are the same size and all corresponding entries are
equal.
Matrix Addition and Subtraction
• Add or subtract corresponding elements.
• The matrices must be of equal size.
• The order does not matter for addition.
Transpose of a Matrix
• For a matrix M of size n×m, the transpose is M T of size m×n.
• Switch the rows and columns. The entries along the diagonal do not change.
Scalar Multiplication
• A scalar is just an ordinary number, as opposed to a matrix.
• Each entry in the matrix is multiplied by the scalar.
Section 2.5
Matrix Multiplication
• Order matters!
o In general, AB ! BA for matrices A and B.
• Compare the dimensions of the matrices you want to multiply.
o The "inside dimensions" have to be the same, or the product is undefined.
o The "outside dimensions" provide the dimensions of the product.
o In other words, if A has size m × n and B has size r × s, then n must equal
r and the resulting product matrix will have size m × s.
• Multiply row by column.
© Zarestky
Math 141 Week in Review
Identity Matrix
• 1’s on the diagonal and zeros elsewhere.
• Always a square matrix.
• Size represented by a subscript: I n is an identity matrix of size n × n.
• Any matrix multiplied by the identity matrix does not change.
Matrix Multiplication and Systems of Linear Equations
• A system of linear equations can be written as a product of matrices, called a matrix
equation.
• You need one matrix for the coefficients, A, one for the variables, X, and one for the
right hand side, B. Then AX = B .
Section 2.6
Inverse of a matrix
• A matrix multiplied by its inverse equals I: A !1 A = AA !1 = I
• A matrix must be square in order to have an inverse.
• A matrix with no inverse is called singular.
• Use the inverse to solve a matrix equation: X = A!1B
Section 2.7
Leontief Input-Output Model
• Use matrices to represent the relationship between production and consumption.
• Assume everything is consumed, and that supply always equals demand.
• Requires 3 matrices:
o A, the input-output matrix. It gives the required input to produce 1 unit of
output.
o X, the gross production matrix. It gives the total output of the economy.
o D, the external (consumer) demand matrix
• AX is the internal consumption matrix. It tells you how much was consumed in the
internal process of production in order to meet the gross output.
• X − AX = D. (total output − internal consumption = external demand). The solution
to this matrix equation is X = (I − A)−1D.