Matrices
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| matrix/ˈmeɪtrɪks/ | ma trận |
| scalar/ˈskeɪlə/ | thực số (bậc) |
| matrix multiplication/ˈmeɪtrɪks ˌmʌltɪplɪˈkeɪʃn/ | phép nhân ma trận |
Numbers in a grid
- Sometimes one number is not enough — you have a whole table of them.
- A price list, a set of coordinates, a system of equations: all are grids of numbers.
- A matrix is that grid, treated as a single mathematical object.
- Once packaged this way, whole tables can be added, scaled, and multiplied at once.
What a matrix is
- A matrix 矩阵 is a rectangular grid of numbers arranged in rows and columns.
- Its size is written rows × columns: $\begin{bmatrix}1&2\\3&4\end{bmatrix}$ is a $2 \times 2$ matrix.
- Each number inside is an entry, located by its row and column.
- A matrix with one row or one column behaves like a vector.
A matrix is…
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A matrix arranges numbers in a rectangular grid; its size is given as rows × columns.
A matrix with 2 rows and 3 columns is called a 2 × ____ matrix.
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Dimensions are always rows first, then columns: 2 rows, 3 columns is a "2 × 3" matrix.
Adding and scaling
- Add two matrices of the same size by adding matching entries.
- $\begin{bmatrix}1&2\\3&4\end{bmatrix} + \begin{bmatrix}5&0\\1&1\end{bmatrix} = \begin{bmatrix}6&2\\4&5\end{bmatrix}$.
- Multiply a matrix by a scalar 标量 by scaling every entry: $2\begin{bmatrix}1&2\\3&4\end{bmatrix} = \begin{bmatrix}2&4\\6&8\end{bmatrix}$.
- These work entry by entry, just like with ordinary numbers.
To multiply a matrix by a scalar, you…
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Scalar multiplication scales every entry: $2\begin{bmatrix}1&2\\3&4\end{bmatrix} = \begin{bmatrix}2&4\\6&8\end{bmatrix}$.
Select all true statements about matrices.
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A matrix holds many numbers, not one. The other three are correct.
Matrix multiplication
- Matrix multiplication 矩阵乘法 is different: it combines rows with columns.
- Each entry of the product pairs a row of the first with a column of the second, multiplying and adding.
- The number of columns in the first must equal the number of rows in the second.
- Order matters: $AB$ is usually not the same as $BA$.
A matrix transforms the plane
A 2x2 matrix moves every point of the plane; this one shears it sideways.
Matrix multiplication multiplies matching entries of the two matrices.
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False. Each entry of the product combines a row of the first matrix with a column of the second (row-by-column dot products), not matching entries.
Matrix multiplication is not entry-by-entry. To find one entry of $AB$, take a whole row of $A$ and a whole column of $B$, multiply matching numbers, and add. Also, the sizes must line up: (columns of $A$) $=$ (rows of $B$), or the product does not exist.
Row-by-column, one entry at a time:
- Top entry: row $(1, 2)$ with column $(5, 6)$ → $5 + 12 = 17$.
- Bottom entry: row $(3, 4)$ with column $(5, 6)$ → $15 + 24 = 39$.
A matrix is a rectangular grid of numbers, sized rows × columns. Add same-size matrices entry by entry; a scalar scales every entry. Matrix multiplication is row-by-column (not entry-by-entry), needs the inner sizes to match, and depends on order.
Write from a top-left entry of 6
- Add the matrices with rows (1, 2), (3, 4) and rows (5, 0), (1, 1).
- Give the top-left entry of the sum.
Add the matrix with rows (1, 2) and (3, 4) to the matrix with rows (5, 0) and (1, 1). What is the top-left entry of the sum?
1 plus 5 is 6.
What is the top-left entry?
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Entry-by-entry addition gives 6.