1. Introduction

Matrix multiplication is a fundamental operation in linear algebra. It is different from element-wise multiplication. In this guide, we'll walk through the Swift code to multiply two matrices.

2. Program Overview

Our approach will be:

1. Define two matrices.

2. Use a function to multiply the matrices.

3. Print the resulting matrix.

3. Code Program

``````// Function to multiply two matrices
func multiplyMatrices(_ matrixA: [[Int]], _ matrixB: [[Int]]) -> [[Int]]? {
let aRows = matrixA.count
let aCols = matrixA[0].count
let bRows = matrixB.count
let bCols = matrixB[0].count

// Ensure matrices can be multiplied
guard aCols == bRows else {
return nil
}

var result = [[Int]](repeating: [Int](repeating: 0, count: bCols), count: aRows)

for i in 0..<aRows {
for j in 0..<bCols {
for k in 0..<aCols {
result[i][j] += matrixA[i][k] * matrixB[k][j]
}
}
}

return result
}

// Define two matrices for testing
let matrixA = [    [1, 2, 3],
[4, 5, 6]
]
let matrixB = [    [7, 8],
[9, 10],
[11, 12]
]

// Multiply the matrices
if let multipliedMatrix = multiplyMatrices(matrixA, matrixB) {
print("Multiplied Matrix:")
for row in multipliedMatrix {
print(row)
}
} else {
print("The matrices cannot be multiplied.")
}
``````

Output:

```Multiplied Matrix:
[58, 64]
[139, 154]
```

4. Step By Step Explanation

1. The function multiplyMatrices(_:_:) is designed to multiply two matrices. It returns an optional 2D integer array.

2. We start by determining the dimensions of the input matrices. Matrix multiplication is possible only when the number of columns in the first matrix (matrixA) equals the number of rows in the second matrix (matrixB).

3. We ensure that matrices can be multiplied by using the guard statement. If they cannot be multiplied, the function returns nil.

4. Nested for loops are used to compute each element in the resulting matrix. The third loop (with index k) performs the summation of the products of the corresponding elements.

5. For testing purposes, we define two matrices (matrixA and matrixB).

6. We then call the multiplyMatrices(_:_:) function and print the multiplied matrix if the matrices can be multiplied.

By following this procedure, we can successfully multiply two matrices in Swift, given they meet the criteria for matrix multiplication.