Principles of Mathematics - Chapter 1: Matrices

True (A)

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

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

<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

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

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

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

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

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

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

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

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

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

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

<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

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

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

A rectangular array of numbers arranged in rows and columns.

A matrix with one row and multiple columns.

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.

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

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.

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

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

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.

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

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

Multiplying each element of a matrix by a constant value.

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

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

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

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.

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.

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

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