To solve the question, we will analyze each matrix provided and determine the following:

1. Bilangan baris (number of rows): Count the number of rows in the matrix.
2. Bilangan lajur (number of columns): Count the number of columns in the matrix.
3. Peringkat (rank of the matrix): Find the rank of each matrix, which is the maximum number of linearly independent rows or columns.

Here is the breakdown for each matrix:

(i) $\begin{pmatrix} 23 \\ -5 \end{pmatrix}$

1. Number of rows (baris): 2.
2. Number of columns (lajur): 1.
3. Rank: As it is a column matrix with two non-zero elements, the rank is $1$.

(ii) $\begin{pmatrix} 2 & 1 & 4 \\ -2 & 6 & 5 \end{pmatrix}$

1. Number of rows (baris): 2.

2. Number of columns (lajur): 3.

3. Rank: Using row reduction:

   • Row 1: $[2 \ 1 \ 4]$
   • Row 2: $[-2 \ 6 \ 5]$

   Perform $R_2 \to R_2 + R_1$:

   2 & 1 & 4 \\ 0 & 7 & 9 \end{pmatrix}$$ Since both rows are independent, the rank is $$2$$.

(iii) $\begin{pmatrix} 7 & 5 \\ -1 & 35 \\ 8 & 1 \end{pmatrix}$

1. Number of rows (baris): 3.

2. Number of columns (lajur): 2.

3. Rank: Using row reduction:

   • Row 1: $[7 \ 5]$
   • Row 2: $[-1 \ 35]$
   • Row 3: $[8 \ 1]$

   Perform Gaussian elimination:

   1. Use the first row as a pivot. Perform $R_2 \to R_2 + \frac{1}{7} R_1$ and $R_3 \to R_3 - \frac{8}{7} R_1$.
   2. Reduce further to check linear independence.

   After reduction, two rows remain independent, so the rank is $2$.

Let me know if you'd like to see the full details of the row reduction or any clarification.

Relative questions:

1. What is the process to determine the rank of a matrix?
2. How can the Gaussian elimination method simplify solving matrix rank problems?
3. What is the difference between a row matrix and a column matrix?
4. How does the number of rows and columns affect the rank of a matrix?
5. Can a rectangular matrix have a rank higher than the smaller dimension (rows or columns)?

Tip: Always check for row or column dependence to determine the rank, as dependent rows/columns reduce the rank.