Principles of Mathematics - Chapter 1: Matrices

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Questions and Answers

What defines the order of a matrix?

  • The largest element in the matrix
  • The number of rows and columns it has (correct)
  • The difference between its rows and columns
  • The sum of its elements

Matrix addition is commutative.

True (A)

What is the identity matrix for a 3 by 3 matrix?

I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

In matrix addition, the resulting matrix C is defined as cij = aij + bij, where A and B have the same dimension. This can only be performed if A and B are of a ____ dimension.

<p>same</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Additive Inverse = A matrix that does not affect the sum when added Identity Matrix = A square matrix with ones on the diagonal and zeros elsewhere Commutative Property = The order of addition does not change the result Associative Property = Grouping of numbers does not affect the sum</p> Signup and view all the answers

What is the result of the matrix multiplication AB where A = [[2, -1], [-1, 0], [2, 1]] and B = [[3, -2], [1, -4]]?

<p>[[0, -7], [10, -9]] (D)</p> Signup and view all the answers

Matrix multiplication is commutative.

<p>False (B)</p> Signup and view all the answers

What is the value of AB if A = [[1, 4], [-2, 5]] and B = [[1], [5]]?

<p>-6</p> Signup and view all the answers

In order for two matrices to be multiplied, the number of ___ in the first matrix must equal the number of ___ in the second matrix.

<p>columns; rows</p> Signup and view all the answers

Match the matrices to their correct multiplication results:

<p>AB = [[0, -7], [10, -9]] BA = [[4, -2], [20, -10]] Matrix commutativity = Not applicable</p> Signup and view all the answers

What is the correct result of the matrix addition A + B if A = [[1, 2, 3], [-2, 1, 4]] and B = [[4, -2], [-2, 4, 4]]?

<p>Not possible due to dimension mismatch (B)</p> Signup and view all the answers

Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix.

<p>True (A)</p> Signup and view all the answers

What do you obtain when you multiply matrix A = [[1, 2, 3], [-2, 1, 4]] by a constant k = 2?

<p>[[2, 4, 6], [-4, 2, 8]]</p> Signup and view all the answers

The resulting matrix from multiplying a 2x3 matrix by a 3x4 matrix is a _____ matrix.

<p>2x4</p> Signup and view all the answers

What is the result of 2A, if A = [[1, 2, 3], [-2, 1, 4]]?

<p>[[2, 4, 6], [-4, 2, 8]] (C)</p> Signup and view all the answers

Match the following operations with their corresponding results:

<p>A + B = [5, 0, 3] A - B = [-3, 4, 8] 2A = [[2, 4, 6], [-4, 2, 8]] 3B = [[12, -6, -15], [-2, 12, 12]]</p> Signup and view all the answers

What is the result of B - A if A = [[1, 2, 3], [-2, 1, 4]] and B = [[4, -2], [-2, 4, 4]]?

<p>[[-3, -4, -8]]</p> Signup and view all the answers

B x A can be performed if A is a 2x3 matrix and B is a 3x2 matrix.

<p>True (A)</p> Signup and view all the answers

Flashcards

Matrix

A rectangular array of numbers arranged in rows and columns.

Vector

A matrix with one row and multiple columns.

Order of a Matrix

The number of rows and columns a matrix has. For example, a matrix with 3 rows and 4 columns has an order of 3 by 4.

Identity Matrix

A matrix where all elements except those on the main diagonal are zero. Elements on the main diagonal are all 1.

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Matrix Addition

Adding corresponding elements of two matrices with the same dimensions. For example, the sum of matrices A and B is obtained by adding A_{11} + B_{11}, A_{12} + B_{12}, and so on.

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Matrix Multiplication

The product of two matrices, where the number of columns in the first matrix must equal the number of rows in the second matrix.

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Matrix Multiplication: Commutativity

The order in which you multiply matrices affects the result. AB does not always equal BA.

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Matrix Multiplication: Element calculation

To find the element at position (i, j) in the product matrix, multiply the i-th row of the first matrix by the j-th column of the second matrix.

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Identity Matrix (I)

A matrix with all elements being zero except for the diagonal, which contains only '1's. It doesn't change the matrix after multiplication.

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Matrix Multiplication: Resulting Dimensions

The dimensions of the resulting matrix after multiplying two matrices. It depends on the dimensions of the original matrices.

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Constant Multiplication of Matrices

Multiplying each element of a matrix by a constant value.

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Addition of Matrices

Addition is only possible when matrices have the same dimensions (number of rows and columns). Add corresponding elements of both matrices.

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Subtraction of Matrices

Subtraction is only possible when matrices have the same dimensions. Subtract corresponding elements of both matrices.

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Scalar Multiplication of a Matrix

The result of multiplying a matrix by a constant is a new matrix with each element multiplied by that constant.

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Matrix Multiplication: Compatibility

In general, multiplication is possible only if the number of columns in the first matrix equals the number of rows in the second matrix. The resulting matrix will have the dimensions of the number of rows in the first matrix and the number of columns in the second.

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Matrix Multiplication Outcome: Dimensions

Think about a matrix with 'm' rows and 'n' columns being multiplied by one with 'n' rows and 'p' columns. The resulting matrix will have 'm' rows and 'p' columns.

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Matrix Multiplication is Not Commutative

Even if A x B is possible, it doesn't necessarily mean B x A will be possible. Matrix multiplication is not commutative.

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Matrix Multiplication: Process

Matrix multiplication involves multiplying and summing elements according to specific rules (like dot products).

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Study Notes

Principles of Mathematics - Chapter 1: Matrices

  • Matrices are arrays of numbers with dimensions m (rows) by n (columns)
  • A vector can be seen as a 1 x m matrix
  • Matrix order, for example, 4 by 3, signifies 4 rows and 3 columns
  • Matrix elements are denoted as aij, where i represents the row and j represents the column
  • An identity matrix has 1's on the main diagonal and 0's elsewhere; it's used to multiply with any matrix and results in the original matrix
  • Matrix addition: Add corresponding elements of two matrices of the same dimension
  • Matrix addition is commutative and associative
  • Matrix elements must be of the same dimension to add them together
  • Matrix multiplication depends on the number of columns in the first matrix and the number of rows in the second matrix

Matrix Operations

  • Multiplication by a constant: Multiply all elements of the matrix by the constant
  • Matrix multiplication is not commutative (AB ≠ BA)
  • Order of matrix multiplication is important
  • Only possible to multiply matrices if the number of columns in the first matrix equals the number of rows in the second matrix

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