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Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only if the number of columns of A equals the number of rows of B, and the number of columns of B equals the number of rows of C (in particular, if one of the products is defined, then the other is also defined).
Then we take slices from V and COL_INDEX starting at row_start and ending at row_end. To extract the row 1 (the second row) of this matrix we set row_start=1 and row_end=2. Then we make the slices V[1:2] = [8] and COL_INDEX[1:2] = [1]. We now know that in row 1 we have one element at column 1 with value 8.
row addition, that is adding a row to another. row multiplication, that is multiplying all entries of a row by a non-zero constant; row switching, that is interchanging two rows of a matrix; These operations are used in several ways, including solving linear equations and finding matrix inverses.
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
These elementary row operations include the multiplication of a row by a nonzero scalar and the addition of a scalar multiple of a row to one of the rows above it. For example: [ 1 3 − 1 0 1 7 ] → add row 2 to row 1 [ 1 4 6 0 1 7 ] . {\displaystyle {\begin{bmatrix}1&3&-1\\0&1&7\\\end{bmatrix}}{\xrightarrow {\text{add row 2 to row 1 ...
The key feature of spreadsheets is the ability for a formula to refer to the contents of other cells, which may, in turn, be the result of a formula. To make such a formula, one replaces a number with a cell reference. For instance, the formula =5*C10 would produce the result of multiplying the value in cell C10 by the number 5.
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The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop: