Solve this question for a class 12 student.---**Question Stem:**
1. In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$,
write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.
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Let's examine the given matrix A. A matrix is a rectangular array of numbers arranged in rows and columns. We use subscript notation a-i-j to reference elements, where i represents the row number and j represents the column number. This matrix contains various types of numbers including integers, fractions, and square roots.
Now let's determine the order of matrix A. The order means the dimensions of the matrix, expressed as rows times columns. Let's count the rows first. We have row 1, row 2, and row 3, so there are 3 rows total. Next, let's count the columns. We have column 1, column 2, column 3, and column 4, so there are 4 columns total. Therefore, the order of matrix A is 3 by 4, written as 3×4.
Now let's count the total number of elements in matrix A. The total number of elements equals rows times columns. Since we established that matrix A has order 3 by 4, we calculate 3 times 4 equals 12 elements. Let's verify this by counting each element one by one, going row by row. Starting from the first row, then second row, and finally the third row. Indeed, we have exactly 12 elements total.
Now let's identify the specific elements using subscript notation. For a-1-3, we look at row 1, column 3, which gives us 19. For a-2-1, we look at row 2, column 1, which is 35. For a-3-3, we look at row 3, column 3, which is negative 5. For a-2-4, we look at row 2, column 4, which is 12. Finally, for a-2-3, we look at row 2, column 3, which is 5 over 2.
Let's summarize our complete solution. First, the order of matrix A is 3 by 4, meaning it has 3 rows and 4 columns. Second, the total number of elements is 12, calculated as 3 times 4. Third, the specific element values are: a-1-3 equals 19, a-2-1 equals 35, a-3-3 equals negative 5, a-2-4 equals 12, and a-2-3 equals 5 over 2. These concepts of matrix dimensions, element counting, and subscript notation are fundamental for working with matrices in mathematics.