Sure! Let's start by understanding the problem and then we will solve it step-by-step.
We are given a matrix and its transpose:
Original matrix:
[tex]\[ \left[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right] \][/tex]
Transpose matrix:
[tex]\[ \left[\begin{array}{cc} 1 & a+1 \\ b+2 & 4 \end{array}\right] \][/tex]
Recall that the transpose of a matrix is obtained by swapping its rows and columns. Therefore, for the given original matrix and its transpose, the corresponding elements must match when the rows and columns have their positions swapped.
Let's represent the elements of the original matrix as:
[tex]\[ \left[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right] \][/tex]
Where:
- The element at the first row and first column is 1.
- The element at the first row and second column is 3.
- The element at the second row and first column is 2.
- The element at the second row and second column is 4.
Now, let's write down what those entries are in the transpose matrix and make the correspondences clear.
The transpose of the original matrix is:
[tex]\[ \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right] \][/tex]
However, our problem gives us the transpose matrix in a slightly modified form:
[tex]\[ \left[\begin{array}{cc} 1 & a+1 \\ b+2 & 4 \end{array}\right] \][/tex]
We need to make the elements in our provided transpose form align with the elements in the true transpose. So let's match each element from our provided transpose matrix to the known values from the actual transpose:
1. The element at the (1, 1) position:
[tex]\[ 1 \][/tex]
2. The element at the (1, 2) position should be the same as the element at the (2, 1) position of the original matrix, which is 2.
[tex]\[ a + 1 = 2 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a + 1 = 2 \][/tex]
[tex]\[ a = 2 - 1 \][/tex]
[tex]\[ a = 1 \][/tex]
3. The element at the (2, 1) position should be the same as the element at the (1, 2) position of the original matrix, which is 3.
[tex]\[ b + 2 = 3 \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b + 2 = 3 \][/tex]
[tex]\[ b = 3 - 2 \][/tex]
[tex]\[ b = 1 \][/tex]
4. The element at the (2, 2) position:
[tex]\[ 4 \][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the transpose matrix match the given conditions are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 1 \][/tex]
To summarize:
[tex]\[ a = 1, \ b = 1 \][/tex]
